First off, this is an extremely vague and difficult question, like robertm pointed out.Can mankind go beyond the limits of spacetime?
But I will try to answer from my own perspective nevertheless.
First off, defining timespace is not that easy. We could define it as everything observable, but this brings numerous issues like the issue about subjective perception.
But let's say for arguments sake we define it as everything we can observe, and by observe I mean touch, see, hear, and also observe in the laboratory like with a hadron collider.
In that case, it gets a bit more easy. For one, humans cannot move their bodies into a dimension or spatial object that is not compatible with our senses and our bodies, so in the physical sense, if we can go there, we can 'transcend' to it.
But that of course brings the issue if we are even transcending, and if it's not just another dimension of reality.
Speaking of transcending too, the word of course implies that we change in order to enter this new reality, which could imply that we could create technology that fits with the new reality, but that our bodies originally do not fit.
If such is the case, the new reality would still need some tangible or other connection to the current reality, otherwise /nothing/ from our reality could enter into it.
And lastly, there is the subjective standpoint on the question.
The problem of awareness is that once humans are aware of something, we automatically get a choice.
So in the ultimate sense, if we become aware of our own inner workings, we should in theory be able to make a choice outside of physical reality.
Right now we are not aware on any level the inner workings of our brain, and most likely this is impossible, but if it isn't, maybe we can 'transcend' our brain, and become aware on a much deeper level, but that is only if the mind is capable of altering matter, but this is too complicated of an issue to get into here, and most likely it's not true either way.