I have also wondered about this question and recently came across some papers that seem to answer it.

First of all, the paper

- Manuel L. Reyes,
*Obstructing extensions of the functor Spec to noncommutative rings*, Israel J. Math. 192 (2012), no. 2, 667–698, arXiv.

shows that there are problems with extending the Spec functor from commutative rings to noncommutative rings.
In particular, any functor Spec : Rings -> Spaces whose restriction to CRings coincides with the usual spectrum functor must map the matrix algebras $\mathrm{M}_n(\mathbf{C})$ ($n \ge 3$) to the empty space.

One can still hope to have a spectrum in the category of locales or toposes.
However, the following paper shows the same result for functors valued in either of these categories.

- Benno van den Berg and Chris Heunen,
*Extending obstructions to noncommutative functorial spectra*, 2012, pdf.

Finally, one can try to represent noncommutative rings by sheaves on Rings with respect to some subcanonical Grothendieck topology that extends the Zariski topology.
This also fails, as described in the paper

- Manuel L. Reyes,
*Sheaves that fail to represent matrix rings*, 2014, arXiv.

However this approach can be made to work, after modifying the notion of descent.
For this, Rosenberg has developed the notion of a Q-category.
He realized that, though there is no suitable Grothendieck topology on Ring, it does admit the structure of a Q-category.
See, for example,

- Maxim Kontsevich, Alexander Rosenberg,
*Noncommutative spaces*, 2004, pdf.

or various other preprints of Alexander Rosenberg.

Interestingly, the work of Rosenberg predates the above "no-go" results by about a decade.

definedas rings: one started with useful geometric objects, and it took Grothendieck's genius to realize that they are the same as rings. What are the useful geometric objects behind your definition? – Alex Degtyarev Mar 5 at 22:36