I have checked a calculator website which checks if a set of vectors is a basis and the one that I put in the title is not. (http://www.mathforyou.net/en/online/vectors/basis/)
By the subspace definition, the vector 0 is contained into the span, 0*(1,0,0,0,0)+0*(0,1,0,0,0), and we can get any other vector from the span with just the independent vectors (1,0,0,0,0) and (0,1,0,0,0) and this span must be a subspace and basis too.
Please tell me if I m doing anything wrong or I misunderstand any concept.
The span of those vectors is indeed a subspace of dimension $2$ of $\mathbb R^5$, whose dimension is $5$. So, those two vectors do not form a basis of $\mathbb R^5$. That's all.