First. What would be a realistic crew count for a spaceship with no independent FTL capabilities? Specifically, in the current timeframe of the universe in question, humanity's FTL capabilities are limited to fixed gates with a travel speed of 99.187856c (roughly 0.8857 parsec per day), requiring weeks of travel time between inhabited systems. The gates utilize hyperspatial wormholes to chuck ships at each other; the targeting is not exact, but the gates are capable of pulling in wormholes passing in their general vicinity and opening a safe exit aperture. The range for doing so is sufficient that one gate per system is enough (and as of right now, the targeting is not accurate enough to pick between multiple gates in the same system anyway, that's part of a later upgrade).
However, the gates have a very low tolerance for gravity: if the gate is located too close to a gravity well like a planet, the gravity-induced curvature of space as described by general relativity pulls the exit aperture away from the gate (the gate pulls in the wormhole with an artificial version of the same effect, but at such close proximity, a planet yanks much harder than a gate) to the point where the wormhole either misses the gate and never opens an exit aperture, opens the aperture somewhere in interstellar space, or opens the exit aperture inside the planet with predictable results for the ship.
Because of this, gates are usually deployed far above or below the system ecliptic, though there is one loophole for this: L1 Lagrange points in two-body systems where the mass difference between the bodies in question is below a certain limit and there isn't a third, much more massive body nearby are stable enough for a gate to operate there. The Earth/Moon L1 is one such location (within tolerance and Jupiter is too far away to interfere) but if there isn't such a convenient spot available, ships have no choice but to spend weeks traveling on laser-ignition fusion rockets between the local worlds and the local gate on top of the weeks they already spent in FTL.
So then. With that kind of travel time, what should be the expected crew size for military vessels if we account for how much storage space would be available for provisions on the following size classes, also accounting for space taken up by ordinance (none are using energy weapons):
- Frigate-class: 96 m, AA and picket - 24 x AA autocannons, 1 x 90mm spinal coilgun
- Destroyer-class: 325 m, medium-to-long-range ship-to-ship and orbital fire support - 4 x gun turrets, 300 x cruise missiles in a broadside version of VLS, 1 x 155mm spinal coilgun
- Carrier-class: 743 m, force projection and taskgroup-level C&C - 32 x AA autocannons, hangar space for 30 fighters, 15 bombers and 20 transports (same chassis as a bomber)
- Battlecruiser-class: 968 m, taskgroup defense (that is, packing an assload of armor to park in front of smaller friendlies and literally take some for the team while said team is returning it with interest), strategic nuclear fire support - 4 x heavy turrets, 18 x medium turrets, 2 x 155mm spinal coilguns, classified number of nuclear-tipped cruise missiles
For force organization, presume that one destroyer and four frigates counts as a task group, one battlecruiser, one carrier and two task groups count as a carrier group. 100+ km ranges are considered standoff because of projectile speeds; shootouts usually don't close to visual range unless a carrier is involved. Due to gates being the sole method of FTL (as of now) humans have, interstellar offensives tend to be very messy and costly because the incoming fleet is naturally chokepointed at the local gate, requiring them to come through very tightly packed to bring enough force to bear quickly enough to avoid being picked off one by one as they arrive.
And before you go "that's fucking pathetic", that's kinda the point: humanity is very new to this whole space warfare thing. The first time the fleet has to fight for real against a force with superior ships and far superior numbers, they suffer a catastrophic asskicking: the humans bring three quarters of their entire navy, slightly over a thousand ships not counting strike craft; the opposition brings a million-plus fleet, each individually superior to every human ship class, roll over the humans with barely any effort and strike a blow humanity doesn't recover from for millennia.
Second. Accounting for the above travel times, if the government suddenly started mass-relocation of people to offworld colonies to scatter humanity so that no single calamity can render them extinct, realistically how many they could relocate within about 15 years tops if they went from zero space infrastructure to interstellar with at least half a dozen settled systems within 23 years? As an added assistance, the government in question has no rivals aside from permanent insurgencies in the Middle East and Central Africa, isn't democratic enough to not resort to military intervention to crush attempted secessions and isn't naive enough to not have near-omnipotent political officers keeping the military in line. Would somewhere in the low eight-digit range too much to ask?
Note that if it weren't for this relocation program, humanity would've went extinct in the aforementioned asskicking, right then and there. Even so, they promptly degenerate into civil war and start killing each other despite their diminished numbers until they're too exhausted to fight and decide to call it off for now until some power consolidation can be had.
Also note that not a single settled planet has a breathable atmosphere or liquid surface water and only about 2 or 3 are even Goldilocks planets. The one planet that's closest to Earth in terms of habitability (similar climate and atmospheric pressure, 1.1g gravity, N-He-CO2 atmosphere, 52 hour rotation) is the most heavily populated and industrialized colony, recently started outlining plans for a space elevator and is under terraforming (icy asteroids from the local asteroid belt have been redirected at the planet to seed it with water and the lakes that have formed have been seeded with algae) but is not expected to be fully habitable for centuries, if not more. Another planet with water, oxygen and even intelligent native life is discovered later, but is uninhabitable to humans due to lethally high oxygen partial pressure and crushing gravity that requires a powered exoskeleton to even walk around in and is dangerously close to the maximum sustained gee tolerance of a human anyway.