Let C be a compact convex subset of 3-dimensional Euclidean space E(3) whose interior is non-empty and whose diameter is d. What is the largest volume that C can have if every subset of C that is a straight line segment of length d is a subset of the boundary of C (or-equivalently-if no chord of C that contains an interior point of C has length d)? It looks as though this maximum might be close to the volume of a right circular cone whose base radius is d/2 and whose height is ((3^(1/2))/2)*d. I am unable to say more. Perhaps there is no maximum volume-only a least upper bound. Although C is a convex body, it is not what would be called stricly convex. To keep things simple I am avoiding any discussion of higher dimensional versions of this question.
I would suggest yet another larger set. Take two points $A$ and $B$ in the plane with $|AB|=d$. Let $S$ be the set of all points $X$ in this plane lying in one halfplane defined by $AB$ and satisfying $|XA|,|XB|\leq d$. Then rotate $S$ around the perpendicular bisector of $AB$; you get the convex body containing your cone. Then, after smashing it a bit you get a desired set --- still of larger volume.
It seems you could curve the base of the cone, making it a sector of a sphere of
radius $r > d$ slightly larger than $d$, centered at a point $c$ directly above the
apex of the cone:
(Arrows indicate segments of length $d$ that cannot fit.) If this is correct, the max volume can be approached but not achieved, as you suggest.
Update. And here is Ilya Bogdanov's larger body, a half-lune: