Firstly, since any vector can be written as a superposition in any given basis, you're always in a superposition of... something.
But OK, I know you were specifically speaking about the position basis.
The continuous interaction between you (as a quantum object) and your environment leads to immediate decoherence, so that position eigenstates become incoherent with each other. So, as you stated, you end up in a very, very narrow superposition of position eigenstates.
However, the magnitude of position dispersion (quantum indeterminacy) is so small than it's drowned inside ordinary (classical) uncertainty.Therefore it's impossible to detect and has no noticeable consequence (as far as we know, of course).
Edit: I didn't want to get too technical, but it's worth mentioning that there are in fact two things happening here.
- Which basis is selected due to interaction with the environment: this process is called decoherence.
- In this basis, which vector is selected by a measurement: this is much more complicated and not completely understood. Let's just say that the wavefunction collapses more or less quickly after a measurement.
In my answer above, I also assumed that a human being is subjected to extremely frequent "measurements", which keeps its wavefunction very close to a position eigenstate.