Here are the things I considered about this question:
1. On a 5x5 board, with perfect play, black can always kill white. Black takes the center point and then any move white plays is basically a futile invasion of black's board. So a 5x5 area definitely is not large enough. This has been proven and the board-size is declared solved. (Ideal komi then is 24 for territory scoring and 25 for area scoring)
2. A 6x6 board does allow both players to live with optimal play and fair komi is currently set at 4. http://senseis.xmp.net/?6x6 So the area does need to be at least 6x6 for the invader to live assuming we are only dealing with edges of the board on the outside.
3. If you invade on the 3-3 point inside a 4-4 with no other stones in the area then your opponent cannot stop you from making unconditional life (although with the double hane variation they can make it small life). This means that with moves that threaten other things (like making life on a side vs. in the corner) can allow a player to live in a relatively small area. This, of course, assumes that you do so on an edge.
So it seems that edges are a very important factor.
4. Making life in the center is really hard, but seems possible in the Shape Game mentioned earlier (my club plays this where the entire edge is considered to be black stones and everyone still lives in the end). With perfect play perhaps it is not because black would seem to control all ladders, but it also seems that white should be able to double-threaten making an eye at least twice.
5. In professional play, players are able to make life in very small areas, even in the center. However, they are not always successful. It would seem that professional play (pretty much the best thing we have to go on) would imply that making life straight up in the center is possible, but incredibly challenging.
6. Pros almost always live in the center by making double threats against walls or by settling for a ko. Whether you poke at a wall or settle for a ko, this means that the outside of the board absolutely matters.
What is on the outside seems to make a lot of difference.
So, I would argue this:
1. It is easiest to invade a corner because the edges seem to help make life. But could you play and live in a 6x6 corner that looked like this? Probably not. Black can just try to force white into a box-4 corner.
If white cannot live here then that would imply that at least a 7x7 area is required, and that is just in the corner!
2. The side is the next easiest to invade, but that would need to be an even larger area. Perhaps as large as 9x9!
This looks hard. Maybe white can later find enough weaknesses in black's position, but I don't think I'd take white in this fight if I had the choice.
3. For living in the center the area must either be very large or you must be able to exploit some type of weakness (be it direct weakness or a ko threat). For that, the Shape Game mentioned earlier would be the thing to analyze. You may need a 17x17 space to make this work unconditionally.
EDIT: I tried this 9x9 living situation and it looks like success may not be possible. I tried myself and failed then had 4 other programs take a crack at it. GnuGo gave up and didn't even try. Pachi tried and quickly failed. MoGo really failed (it couldn't analyze the position correctly). And in two attempts Fuego failed to live (did two with Fuego since MoGo didn't work and Fuego seemed the closest). Here is the SGF of those tries -
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File comment: Attempts by myself and bots to live in a 9x9 area of black influence that may in fact be solid territory.
WhiteToPlayAndLive.sgf [2.8 KiB]
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). One thing that was cautioned was that you need to consider whether there are forcing moves inside this space. If there are then the invader may need less space to live if they are sealed in since they may get free moves that allow them to settle. 