Not really, no, but I can see why you might think that, because Bell's theorem is often described as saying that quantum mechanics contradicts "local realism", but "realism" in this sense has a precise technical meaning, which is that a physical system at some time has a complete set of well-defined values for all the possible measurements that we might do on it (and "local" just means that physical effects can't propagate faster than the speed of light). It was known since the beginning that quantum mechanics, as a theory, doesn't have this feature because of complementary observables, aka the uncertainty principle, which says that the more precisely some quantities are known, the less precisely the theory determines other quantities, e.g. in the case of a spin-1/2 particle, knowing the value of its spin along one axis means that it's value along any axis at right-angles is completely uncertain - it can be up or down with equal probability.

Nevertheless, it was thought (e.g. in the Einstein Podolsky Rosen paper in 1935) that it might be possible to formulate a theory that could reproduce all the correct predictions of quantum mechanics, while also ascribing simultaneous well-defined values to all the physical quantities possessed by a quantum system, i.e a locally realistic theory. These are also known as local "hidden variable" theories, where the idea was that some of the values of the variables might be unobserved simply because of measurement practicalities - we can't measure the spin of a particle along two orthogonal axes simultaneously because the measurement needs a magnetic field gradient along the direction we're measuring in.

Bell derived an inequality that any locally realistic theory must satisfy, and showed that quantum mechanics in fact violates this inequality, so no locally realistic theory can reproduce the predictions of quantum mechanics. Alain Aspect and others later implemented Bell's thought experiment in the lab and showed that the physical world obeys quantum mechanics, and so is not describable by a locally realistic theory.

In my view, none of that shows that there is "no objective reality". Rather, it shows that objective reality is as far as we can tell quantum mechanical, and not locally realistic in the sense described above. It's certainly the case that quantum mechanics requires a modification of the classical concepts of reality, i.e. of classical ideas about what a physical system is, but you would only accept that conclusion if you agree that quantum mechanics is telling you something objective about reality... At least according to how I understand those words.

So I think what people really mean when they say quantum mechanics shows there's no objective reality is just that it contradicts classical conceptions of physical systems, which is clearly true but sounds less sexy and mysterious.