Firstly there are 2 Everest base camps, one in Nepal and one in Tibet.
I'm guessing your after the Nepal one.
Both are set up on large flat areas at around 17,000 feet. There should be no problems with actual space. The area is large and flat. You may not get the best pitch but you will be able to fit somewhere.
Couple of other factors relating to this:
- There is only 2 seasons that you can climb to the base camps (March to May and from September to December).
- You will need a permit, these are expensive and limited in number. Maybe this is what the person was talking about?
- There is also talk of forcing people to get guides, though this has yet to be enforced but could be in the near future. This is due to a number of deaths on the trek up by solo travellers recently.
The fee's depend on where your climbing from, for Nepal the fee's have recently been cut. To base camp your talking between $70-$250 depending on when you want to climb.
