Computing distances with approximation less than 2 in general graphs?
As far as I know, there is no published result on computing distances of approximation less than 2 in subquadratic space and sublinear query time. For retrieving approximate distances quickly, you may want to look at results and references in "Faster Algorithms for All-pairs Approximate Shortest Paths" by Baswana and Kavitha (journal version of their FOCS paper has a good review of related work); none of these achieve subquadratic space.

For compactly retrieving approximate distances, you may want to look at the results and references in the above two papers. [As an addition to the answer by Gabor, a word of caution: be careful about the notion of sparsity in above papers -- for approximation $2$, a graph is said to be sparse if $m = o(n^2)$, as you probably know already].

As Sariel pointed out in one of the comments above, a natural lower bound on space for computing distances of approximation less than $2$ is $\Omega(m)$, that is, linear in the size of the graph. If the query time is not bounded, this lower bound can not be improved (trivially, one can use shortest path algorithm by just storing the graph). For constant query time, I know of two lower bounds. First, Patrascu and Roddity had some conditional lower bounds in their FOCS 2010 paper that apply for approximation less than $2$. Second, Sommer et. al. had some lower bounds for extremely sparse graphs. I am not aware of any other (non-trivial) lower bounds.

In terms of upper bounds, the results from above papers do not seem to generalize to approximation less than $2$. We recently made some progress on this problem. The paper should be on ArXiv soon, but if you like, send me an e-mail and I'll be happy to share the paper.

Hope this helps.

~Rachit Agarwal