how to access the hard drive when the mac doesn't start

Hi,

I have the following problem. I started to upgrade my macbook air from Lion to Yosemite. Unfortunately I couldn't finish the update as I didn't have enough free space on my HD. The problem is I couldn't return to the old OS or I could finish the update. And I didn't do a backup before starting the installation. My question is, if it's posable in this phase to access the HD and save the date or it's to late for this.

I am aware that I have to run the recovery of the HD by deleting the HD. In this case alle my date will be lost.

So if anybody know a method to access the HD in this phase of the Mac I'll be more than grateful.