reinstall yosemite without losing any data

Reinstalling OS X Without Erasing the Drive

Boot to the Recovery HD: Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.

Reinstalling OS X Without Erasing the Drive

Repair the Hard Drive and Permissions: Upon startup select Disk Utility from the main menu. Repair the Hard Drive and Permissions as follows.

When the recovery menu appears select Disk Utility and press the Continue button. After Disk Utility loads select the Macintosh HD entry from the the left side list. Click on the First Aid tab, then click on the Repair Disk button. If Disk Utility reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit Disk Utility and return to the main menu.

Reinstall OS X: Select Reinstall OS X and click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.

Alternatively, see:

Reinstall OS X Without Erasing the Drive

Choose the version you have installed now:

OS X Yosemite- Reinstall OS X

OS X Mavericks- Reinstall OS X

OS X Mountain Lion- Reinstall OS X

OS X Lion- Reinstall Mac OS X

Note: You will need an active Internet connection. I suggest using Ethernet

if possible because it is three times faster than wireless.

Back your data up before proceeding; the reinstallation process won't itself erase your data, but a problem may occur.

(126714)

Yes. Reinstalling OS X does not delete anything.

Back up your system anyway, as a precaution against the unexpected.

does the time machine back up the whole hard drive or just the data portion?

The entire source volume is backed up, unless you exclude items in Time Machine Preferences (under "Options").

I was thinking of erasing the hard drive completely and install the fresh copy of OS X, and then restore the data.

You can certainly do that. I recommend you merely reinstall OS X first, and determine if that fixes the problem. Restoring a corrupted system from a Time Machine backup would obviously reinstall the corruption.