You asked a few questions:
Considering C to be almost an obsolete technology now, how will that impact my career?
As has been pointed out in comments, we need to challenge the assumption that C is "almost obsolete" - certainly, there's a large body of work already developed in C, and that work is being maintained and expanded constantly. In addition, C and C++ have advantages that many "newer" languages don't have, and organizations are continuing to use these "older" languages for new development. In other words, there is absolutely no shortage of work in C.
Secondly, you asked,
Will I be later able to shift if I join this company?
Yes - As mentioned above, there is no shortage of work in C. Further, and perhaps more importantly, learning a fundamental low level language like C is a great way to develop skills that will be easily transferable to just about any development job, even if you decide you don't want to actually develop in the language you've learned. Keep in mind: often, employers are looking for fundamental skills and the ability to learn and solve problems versus expertise in a specific product or language.
Finally, you asked,
What should be my correct choice now?
We can't really answer that, because you need to decide for yourself what the most important factors are.
All that said, I think it's very important to focus on some slightly incorrect assumptions you've made. There seems to be a common thought among new developers that anything old in tech is inherently bad and no one wants it. While it can seem exciting to always work in the most recent language or with the most bleeding edge platform, focusing too much on the newest thing will seriously restrict your applicability to many programming jobs. There will always be a churn of new languages and technologies, but the majority of our technology infrastructure is built on "old" ideas and languages - which, from an employment perspective, means that there will just about always be demand for things many newcomers would label as outdated.