Fictional units of measurement

Real life vs Fiction

The two main approaches to using units of measurement are using real life units and making up your own. Within these two, real life can be split up into modern and historical units, whereas made up ones can be split into 'based on real life' and 'made up words'. To illustrate:
Real, modern: Kilograms, meters, liters
Real, historical: Grains, furlongs, barrels
Fictional, based on real life: Ingots, staves, buckets
Fictional, made up: Dilrets, Gwinnels, Dops

Based on your personal preference you probably already have favorites among these examples, but before you go wild with creating your own units, consider how the names of the units came to be and how each unit could be named in an organic way within your story universe. Names generally don't come from nothing after all. Kilo, for example, comes from the Greek word for thousand, and meter comes from the French mètre (also meaning meter), which in turn comes from the Greek for measure. There are often logical ways to come about names for measurements, which is what I'll delve into deeper for each unit type.

Using and avoiding

Which approach is best for you to use depends a lot on your story universe, as well as personal preferences to some degree. Stories set in modern or futuristic periods can easily use modern units or made up ones for nonexistent measurement types. Stories within medieval-like periods often work best with historical ones, but fictional ones based on real units work great, too. The latter works especially well for fantasy universes trying to separate themselves from real life without bombarding the reader with countless invented words.

Another option, one many writers use, is to simply avoid using units as much as possible. Simply using descriptions is often enough and allows you to keep a distance between real life and your fictional world all the same. For example, instead of "He was about 1 meter 70 (5'7") tall." you could write "He stood shorter than the average man.". The same can be done for almost everything where the measurement itself isn't what's important, but the context behind it.

Before I delve into specific units and ways to name them, I will first advice against creating your own words for common units. There are exceptions, mentioned later, but in general you want to make sure the units you use are familiar to the reader even if they might not make the most sense within your story universe. Some argue it makes sense to use the words the people within the universe use, but the reader doesn't live within your universe the same way the characters do. Plus, you're writing in an Earth language for your reader, so in a way you translate whichever words your story universe characters use to the language you're writing in, including their units of measurement.

Lengths and sizes

Possibly the most commonly used units, it's also one of the easier ones to invent names for. In real life we often use body parts (feet, hands, heads, etc.) or animals (the size of a bear, cow, etc.) or established lengths (football field, light-year, etc.). So if you're not going for real life versions, simply using synonyms or story universe equivalents works great and can add a great layer of world building.
If you're going to invent your own, I'd stick to names based on real life regardless of the setting of your story. It's far easier to envision possible lengths of invented units like canes, staves, rods or stretches than it is to envision and remember the lengths of a made up unit names like Grenchens, Twirrens and Bilts.
If you're not into inventing your own names, I recommend simply going with the real life equivalents that best reflect the setting of your story universe. So modern or futuristic settings paired with modern units, medieval-esque settings with medieval units, ancient world types of settings with ancient world units, and so on

When it comes to inventing systems, using elements within your world that are readily available will go a great way to create a realistic system. It's why we use(d) body parts so often, they're always at hand (ba dum tsh) to measure stuff with.
You could add world building elements to this, too. For example, a banner might be required by law to be 2 meters, which means a banner could also be used as a measurement. 4.5 meters would be 2.25 banners.
Using averages or similar methods of standardization to invent a unit work well, too. Everybody obviously has different feet, but feet became a standard unit regardless.

Mass

Mass is both a fun and tricky one to mess around with in terms of coming up with your own units. Depending on how specific you want to go, it can become difficult to keep things organized in a way that doesn't confuse the reader. In real life we have units like kilos and grams, pounds and ounces, and stones. But our systems aren't always the most intuitive or detailed, so they don't help much when creating our own fictional systems, especially when our new systems have a wider range of orders of magnitude, each in need of a name.
For example, say you wanted a named unit for 0.5 lbs, 1 lbs, 5 lbs, 10 lbs and 100 lbs. You could go for something like pebbles, cobbles, stones, slabs and boulders, but they won't necessarily match the weights you're trying to fit within your system.

To make the process easier, find common objects within your world that could be used as standard units and design your mass unit system around them. Similar to how body parts can dictate units of length. Alternatively, create smaller magnitudes the same way we use grams, kilograms and micrograms.
If your story is set in a medieval world, you could use ingots or coins of different metals, the average weight of a chicken, the average height of a bag of salt, bales of hay or cotton, and so on.

As far as stories set in the future go, sticking to one of our current systems and building from there can be very helpful to both you and your readers. If you're using the metric system, there are already many orders of magnitude available from attograms and yoctograms to petagrams and deltagrams. But the average reader doesn't know which of these is bigger, let alone which one I made up, so unless scientific accuracy is important within your story, it can help to invent names more easily distinguishable.
Here it can again help to use recognizable objects or celestial bodies depending on how large of units of mass you're going for. You could go for Earth, Mega-Earths and Half-Earths, for example, as well as Suns, Mega-Suns, Half-Suns, Moons, Half-Moons and so on.

Volume

Volume is similar to length in terms of approach, but it's more difficult to create new units because there are only so many containers you can use. Barrels, flasks, mugs and thimbles all work, but it becomes more difficult for larger volumes. You could use lakes and ponds and seas, but they're difficult to standardize.
For lakes, ponds, motes and other large volumes you could use the volume of the hulls of ships or the balloons of zeppelins. For seas and oceans it becomes more difficult, but it's unlikely these matter in historical contexts anyway, the technology won't be good enough to measure these beyond perhaps the time it takes to cross them. But if it is important, picking the size of a large lake could work as a unit for your story universe.
I'll illustrate it with an example. Let's say our story universe has two units for large volumes: trade hulls (or T-Hulls) and battle hulls (or B-Hulls). Trade vessel hulls are smaller, let's say half the size of a batteship's hull. A nearby lake, we'll call it Lake Durniss, has a volume of 25 B-Hulls. The nearby sea could have a volume of 50 lakes or 50 Durniss.

As you can see, it becomes difficult to invent names for large volumes without relying in part on making up names. This is where I'd rely on simply avoiding units all together and describing the size instead. As long as the exact measurements are not important to the story, this is the better way to go about it. Descriptions often say a lot more than numbers do, as it's harder to get a true sense of large scopes with just numbers.

Time

Time is a fun one to mess around with. There are many ways to go about it by simply using natural phenomena. Moon cycles, day lengths, equinox and solstices, time measuring mechanisms, and so on. Units of time are especially great for world building, as they're inherently tied to astrology, calendars, time keeping and other day to day tasks we may take for granted. Many of these elements can also show the grasp the people in your world have on science and their position within the world, as well as what they hold important (moon cycles? harvest cycles?).

Taking a look at historical time measuring techniques can lead to easy ways of inventing your own system or your own unit names. Candles, hourglasses, sundials, water clocks, pendulums and so on can all offer names for time units ranging from wicks to shadows, from pops to drops and from clicks to gears.

When it comes to longer time scales, using celestial bodies can work, assuming the people in your story universe have a strong enough grasp of how their universe works that is. Take our solar system as an example, the orbits of the planets range from 88 Earth days to almost 165 Earth years. So Neptune years could be a unit if you so wish.
On the truly large time scales you could go for galaxy rotation times, lifetime of a specific star, and so on. This way you could end up with names like Sundeaths, Gal-spins, and so on.

Because long time spans are often related to advanced technology and science, using made up names is a lot easier. Science is full of scientists' names for all sorts of units (Watt, Ampère, Ohm, etc.), so when it comes to advanced scientific units, using scientist character names is a realistic way of inventing names.

Miscellaneous

There are, of course, far more types of units beyond the previous 4 types. Different forces, economic measurements, biological processes, and a whole bunch more could all come into play in your story and potentially be in need of a new name. Inventing their names and potentially their systems of magnitude can be difficult, but I would go about by asking the following questions:
- Can the unit(s) be related to real life objects?
- Is it advanced science-esque in nature or something anybody could establish?
- Are there real life and potentially historical examples?

With these three examples and the setting context of your story universe (modern, futuristic, etc.) in mind, you can figure out at least a naming system type from the 4 types mentioned at the start.
If the answer to question one is yes, using real life objects can be a great way to keep things realistic and organic, as well as easier for readers.
If the answer to the second question is that your unit(s) is indeed advanced science-esque in nature, you can get away with inventing new words/scientist names. It happens a lot in time travel movies and other stories where the science doesn't actually make any real sense in the real world.
If the answer to question three is yes, using those examples can offer a simple solution and could be the best way forward when this part of the world building is not that important to the reader. Sometimes simplicity is better than trying to explain a new system to a reader. The context of your story matters in deciding this.

Invented words

If you do want to go the way of inventing words, consider approaching it from a linguistic point of view. Just as kilo comes from thousand and meter comes from measure, so too can your units come from words within your fictional languages regardless of whether you've fleshed them out or not. It obviously works better if you do have a fleshed out language ready to be used, but you could use real languages to help you out as well. A hundred in Greek is εκατό (ekató), so your fictional version of a hundred mass units could be ekatos or ekats, for example.
Alternatively, going the scientist name route is always an option as well. It could lead to world building from a different angle and doesn't require a linguistic approach. A mixture is obviously also an option. In the end it's your story universe, so you make the rules.