I am thinking about getting the Panasonic 53" TV. However, the only place to put the TV is in my bedroom which would mean I would be sitting a little over 5' away from it. Is that too close?
That's way too close, if you ask me, although somebody else around here might sit that close ratio-wise (screen-size-to-distance). I have that TV also and use ~8' eyes-to-screen distance. Most people around here seem to use somewhere around 2-2.5x size-to-distance ratio although plenty go higher while a few(?) go lower.
Why not get the 47" instead? You might find it available for much less than the 53" also. And even a 47" might be too big for that distance. If it were me, I'd probably go w/ a 42" or so.
Anyway, only you can determine what's the appropriate viewing distance. Try checking out in person for yourself. It doesn't even have to be that exact model--just adjust for the size-to-distance ratio, if needed. Make sure to see if the PQ for your regular viewing is acceptable at your distance.
I recently saw some fairly convincing math based on the average human eye's ability to distinguish pixels once they are more than 1 minute (1/60th of a degree) of vertical arc apart from one another in the field of vision.
By that math, if you're viewing non line-doubled NTSC on a 16x9 screen, the optimal distance would have the screen occupying about a 20 degree wide arc in your line of sight. For a 53" diagonal, that works out to sitting about 10 feet away. For line-doubled material it works out to about 5 feet away. For 1080i material, it works out to about 42 inches.
Note that this is _strictly_ focused on perceiving as much detail from the picture as possible without getting to a point where it looks line-structured. It takes no consideration of source quality, which will almost definitely be more of a limiting factor if you're watching (for example) non HD cable. Also, the foundation of all of this math is a biological statistic that I cannot validate.
All of this said, I think the previous suggestion to go to a store, sit 5 feet away and watch TV for 5 or 10 minutes is an excellent one. 5 feet sounds too close to me for most source material. Be sure you know what you're watching, though. HD Net at the store will look a LOT better from 5 feet than Cartoon Network would at home....
Here's a link to check out. It's a good calculator for the "optimal" viewing distances based on your screen size (or optimal screen size based on your viewing distance). It uses the SMPTE and THX recommendations (and explains all of it).