With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. 1
For 3-space, add the restriction of no 4 points in a plane. One way to get 5 points at integral distances is to first find a set of 5 Soddy spheres that have integral curvature (curvature=1/radius). For example: $\{1, 3, 4, 9, 16\}$. This set satisfies $3(1^2+3^2+4^2+9^2+16^2)=(1+3+4+9+16)^2$, so it's a set of Soddy spheres. That corresponds to five touching spheres with radii $\{144, 48, 36, 16, 9\}$. The ten distances between points are all integer values.
Is there a set of 5 points with smaller distances?
Are there solutions for 6 points or more?
Change the restrictions, and demand only that the convex hull of points has a volume>0. Here's a set of nine points where all distances have integer value:
What are the smallest possible hull volumes for $K_6$, $K_7$, $K_8$, $K_9$, and beyond?
1 Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021) labeled diagram