Well, most of the resources that are renewable and portable you will want nearby. That means using your shovel to dig up bushes, saplings, and tufts of grass and planting them all nearby. Plant trees nearby as well. Ideally these things will be close enough you can get to them at night to have something to do before the sun comes back up, but you don't want to catch them on fire.
You can add farm plots nearby. With farm plots as well as berry bushes, you should have a pretty stable food source even without having to leave your base, or at least without having to leave very often.
You'll want to be able to get manure -- this means either building your base near a beefalo herd (good because they don't ever go crazy and try to kill you, but they wander) or a pigman village (good because you'll always know where to find them, bad because sometimes they go crazy and try and kill you, plus you have to give them stuff in return for manure).
You may want to be within range of a few other resources. Pigman villages you can construct, spider eggs you can move. Either of these, if you want them, should be placed a few screens away so you don't have to fight through the night. Placing your base within a comfortable walk of some tallbird eggs might be a good idea.
I placed my base near a beefalo herd (which is now spreading out pretty well, so this is becoming less effective) as well as a bunch of rabbit holes. I don't have easy access to tallbird eggs, but between farms, bushes, rabbits, and enough manure to keep my farms and bushes going, I pretty much never have to worry about food or basic resources (sticks, wood, grass).
Other than that, just keep things near your base once you can research them -- crock pot, chests, etc. The things you keep closest to your fire should be the things you want to mess with overnight. Hostile (or potentially hostile) things should be kept a few screens away. Everything else can go somewhere in between.
Best Answer
To avoid dying, you need to watch 3 major states:
Additionally, some security against other ways of dying (mob attacks such as hounds, mactusk, deerclops, ...) can lower your chances of dying too.
The good news is: everything you need to keep the 3 states up can be moved: you can plant farms, move berry bushes, grass, twigs, trees, spiders, ...
Resources you'll need to build a base but are not easily movable are manure and stone. (Update 22/07/2016: manure is movable if you build pig houses, pigs give manure if you feed them fruits. Turn the pig into a werepig and drop a stack of 40 berries to get 40 manure fast)
So you can build your base anywhere you like. You can move all vital resources anyway, it only costs manure to fertilise the grass and farms, and stone to build your walls. It helps to have those close!
Conclusion: build your base close to
Additionally:
A good example of a base in my latest game, day 3: