It's an interesting idea. For instance, we know that when Voyager was trapped tens of thousands of light years away, it was going to take the ship about 70 years or so to return to the Alpha Quadrant. So we know there are limitations on how fast a warp-capable ship can go.
Such a mission would probably be very rare, but suppose the Federation wanted to explore areas of the Alpha and Bete Quadrants that are more than 10,000 light years away. I believe it would take a ship at high warp about 10 years to get to that destination, and another 10 years to get back.
So, as Wingsley suggested, you'd have small crews alternating between running the ship and being in stasis. This means that maybe the captain and first officer would be awake for the first and last parts of each voyage to and from their destination. Then you'd have other "shifts," lasting maybe a year, of skeleton crews to maintain the ship over the long voyage. You'd also need a waking crew to deal with contact with other space-faring civilizations they're bound to come across over the course of their long journey.
Such a mission would have to have an exceptional crew to deal with the stress of returning to the Federation more than 20 years later.
Another possibility is sending a small fleet, again, as suggested by Wingsley, to establish a foothold in a distant sector, to colonize and expand the Federation and Starfleet into that sector, perhaps. So you'd need to put in stasis civilan authorities to run a colony, say a governor and other personnel, with the Starfleet's crew's function is to make sure they get there in one piece.
I can imagine the following scenario: as we know, subspace communication is possible over long distances, and is faster than warp speed. So perhaps the Federation makes long-distance communication with a civilization that either wants to join the Federation or make close contact with its representatives.
I also like Timo's idea that sleeper missions may have been launched to explore neighboring dwarf galaxies, very similar to the Kelvans' expedition in By Any Other Name. I can imagine such missions sanctioned as early as the late 22nd century. Of course, the 24th century Galaxy-class would probably be best suited for such long missions.
Or, even more likely, you'd have to construct a special class of ship to deal with the rigors of such long voyages, not to mention sleeping areas for the part of the crew in stasis, like a special "cargo" bay. Another possibility: building the stasis units in every crew members' quarters to minimize space.
Speaking of the Kelvans, if they did settle on that planet in the ep, and established peaceful relations with the Federation, they might give them the technology to increase warp drive efficiency over long voyages. Also, they might share the technology they used to reduce the whole crew of Enterprise to the small cube shapes as a different kind of stasis -- as we saw, the crew was returned from that state. Storing, say, 400 to 1,000 of those cubes would take a lot less space than stasis chambers with fully grown humanoids sleeping in them.
Another thing to consider: Remember that Riker was going to captain the Aries, and it was to go on a deep space mission that would take three years to get there, and presumably, another three to get back after contact was made with the intercepted signal, assuming, as Riker said, there was life there. I can envision for part of the journey, some of the crew put in stasis to reduce the chances of boredom on the way, just in case they enter a region of space where there are no spacefaring civilizations to encounter. Also, that would extend food supplies and other consumable material.
Red Ranger
