Do you have to allocate a citizen to work a resource in order to gain access to it?

In this answer, Fabian said that it's enough to have an improved resource in your cultural border in order to gain access to that resource; while I was under the impression you have to actually allocate a citizen to work on the improved resource in order to get it (whether strategic or luxury), so for example you could never gain access to a resource which is farther than 3 tiles from your closest city.

I seem to remember a case where I improved a far resource and didn't get it, but I'm not so sure anymore. Which one is it, then? Do I have to actively work it or is having it within my borders enough? And if it's the latter case, can I get even further resources by using a culture bomb, which seems to be able to expand my borders even beyond 5 tiles from the nearest city?

No.

It's my experiance that simply having a tile that contains a resource within your borders and with the relevant improvement constructed (and not pillaged) is sufficent to add it to your network (it doesn't even need a road/rail connection like they did in Civ4) - this is for luxuries and strategic resources, things like sheep that just boost tile output obviously have to be worked (in the same way as you have to work a hill to get the benefit of having built a mine there).

And in a recent game I culture bombed my territory outwards in to a neutral desert area, specifically just to obtain an oil field. It was 5 tiles from my nearest city, and a soon as my worker finished building a well upon the tile I had oil available to be used for construction.

The enemy cannot block your access to the resource (say by standing units on that tile) directly, but it is quite a simple task for them to pillage the square in question, which will then cause you to lose the resource until such time as you repair the pillaged improvement.
(With thanks to bwarner for the clarification in comments).

Note that most (all?) resources also grant an output bonus to the tile they are on, and you have to work the tile from a city to get that bonus.