Let me give an answer in the case of positively curved surfaces.
We define an ovaloid as a compact connected surface $S \subset \mathbb{R}^3$ with $K(p) >0$ for all $p \in S$, where $k$ denotes the gaussian curvature. If $S$ is an ovaloid then $S$ is orientable, and by using Hadamard-Stoker theorem it follows that its inner domain is a convex subset of $\mathbb{R}^3$ and that its Gauss map $N \colon S \to \mathbb{S}^2$ is a diffeomorphism. In particular, $S$ is a topological sphere.
The following classical estimate for the diameter of an ovaloid is due to Bonnet. Modern proofs are based on Hopf-Rinow theorem, see for instance
S. Montel, A. Ros, Curves and Surfaces, Theorem 7.45.
Theorem (Bonnet). Let $S \subset \mathbb{R}^3$ be a closed connected surface whose Gaussian curvature $K$ satisfies $\inf_{p \in S} K(p) >0$. Then $S$ is compact, that is $S$ is an ovaloid, and its diameter satisfies $$\textrm{diam} \, S \leq \frac{\pi}{\sqrt{\inf_{p \in S} K(p)} }.$$