Is it really possible to "catch 'em all"?

It depends on what you mean by "catching 'em all on one game." Obviously, during a playthrough of a single game, you cannot catch a wild individual from every single species; for one thing, the games are designed such that some Pokémon cannot be obtained without trading them from another version. However, via trades and transferring from previous generations, you can gather all Pokémon other than:

• Mew
• Celebi
• Jirachi
• Deoxys
• Phione
• Manaphy
• Darkrai
• Shaymin
• Arceus
• Victini
• Zorua
• Zoroark
• Keldeo
• Meloetta
• Genesect

in a single game, and with events (either by participating in them or just getting the Pokémon from others in trades) you can get everything other than Meloetta and Genesect, which have not been released in any events yet. You can gain enough boxes to hold them all by putting at least one Pokémon in each box and then exiting & reentering the storage system until you've unlocked all 24 boxes.

It is entirely possible to catch 'em all. You will need to do a lot of trading, and it may prove difficult to get hold of the Nintendo event Pokemon, especially the ones which haven't been distributed in years like Mew. There are resources online such as http://www.pokebay.com/ where you can source the Pokemon you need to complete your collection.

Good luck to complete your collection!