To start, there's actually a 20-slot bag that you can get pretty much the moment you turn 70. It's called the Tattered Hexcloth Sack, and it's from Zul'Aman. However, do not despair at the Zul'Aman part. You don't need to be tricked out in Tier 4 and Karazhan gear to get it. In fact, you don't need to kill a single monster to get it, if you play your cards right.
The bag is from a quest called Promises, Promises. Getting the quest is simple: Just head up to the Ghostlands and talk to Budd Nedreck near the entrance of Zul'Aman (talking to Griftah first is optional, but it'll get you a few extra silver and some humorous quest dialogue). He'll give you a quest to retrieve his map from Nalorakk, the first boss of Zul'aman. But again, don't despair. The map itself doesn't drop off Nalorakk. It doesn't even drop off any of the nearby trash. Instead, it's a ground spawn right on his platform that can be picked up by an unlimited amount of people, as long as they have the proper quest.
So, once you decide you want the sack, if you can't get a group together to kill Nalorakk, you just have to find someone who has killed him since the last Zul'Aman reset. Once you've found that special person, have them make a raid group and invite you, then zone into Zul'Aman. You should now be in their Zul'aman instance, open front door and all.
From there, head in the gate past the gong, and keep right, going down 2 flights of stairs. You may want to watch out for a patrol of 2 trolls after the second flight, just in case it respawned. They have a very wide patrol range though, so you should have no problem slipping past them once they turn away. You'll want to head right again, into a grassy side area. Once you round the next corner, there will be a passage to the left. Watch out for the bears near that passage, not every group kills them all. Luckily, there will be no need to aggro them to get the map. Once you're through that passage, head up the stairs immediately on your right, then head up the stairs on your left. If you see a gong straight ahead, you've steered yourself correctly, and you are now on Nalorakk's platform. Head to your right, to a small raised platform, and the map itself should be on the a crate there. Grab the map, and head back out. Talk to Budd, and you'll have your bag, quick and easy. The hardest part is finding someone who'll let you use their partially cleared instance.
So now you've started your bag space expansion, what next? For the immediate future, your best bet are Primal Mooncloth Bags, created by Tailors. The problem is, it takes a LOT of work to create them. The 8 pieces of Primal Mooncloth alone will take over a month for some tailors to make (half a Month for Mooncloth specialized tailors). In addition, considering all the Primal Water and Primal Life needed to create the cloth, you'll be paying a hefty price for each piece.
However, if you don't have the gold, you still don't quite have to fall into despair. You'll just have to wait another month or two for 2.4 to hit the live servers, and you'll get two more relatively easy 20-slot bag options. The first is the Sun-Touched Satchel. For this one, you will have to kill a few mobs, as it drops in the Magister's Terrace. It appears it drops in normal mode off any mob though, so hopefully it shouldn't strain your resources too hard to go on a run or two. And hey, you'll probably be wanting to grind some Shattered Sun Offensive reputation in there anyway.
The next one's going to be a bit harder for most people to get, but it's still worth mentioning. Starting in 2.4, Magtheridon's getting a big facelift. Among other things, this comes a new piece of loot, the Pit Lord's Satchel. The great thing about the satchel is that it's non-unique. This means that as long as you can sweet-talk the other 24 members of your raid group, you can conceivably pick up a few of these and not have to worry about farming up Primal Mooncloth materials at all. Also, since Magtheridon's getting a lot easier next patch, a lot more people should have a shot at killing him and grabbing this bag.
So there you have it. We may not quite have bags of holding yet, but we can at least get a bit more space while adventuring Azeroth. So I say to you, go forth and multiply your bag space, and may you never again have to destroy another stack of Crisp Basilisk Urethras because of your dinky little bags.