This is a design theory question.  You are asking about the existence of a [Balanced Incomplete Block Design][1] (BIBD).  A *$(t,v,k,\lambda)$-design* is a collection of $k$-subsets (called *blocks*) of a $v$-element set $X$ such that every $t$-subset of $X$ is contained in exactly $\lambda$ blocks.  You are interested in the case $t=\lambda=2$.  There are some obvious divisibility conditions.  For example, $\binom{k}{2}$ must divide $\lambda \binom{v}{2}$.  Moreover, the condition on pairs also implies that the number of blocks containing a single element is also constant.  So, $k-1$ must also divide $\lambda (v-1)$. It was a famous open problem called the *Existence Conjecture*, whether these divisibility conditions were also sufficent (up to a finite number of counterexamples).  In a series of influential papers in the 1970s, Richard Wilson solved the Existence Conjecture for $t=2$.  Recently, Peter Keevash solved the Existence Conjecture in general.  See the paper [The existence of designs][2] and the references therein.  


  [1]: https://en.wikipedia.org/wiki/Block_design
  [2]: https://arxiv.org/abs/1401.3665