HST is in Low Earth Orbit. It can of course not look through Earth which at that low altitude covers almost half of the sky. As far as I know it is also never pointed even remotely close to the direction of the Sun for safety reasons. When the HST is in between the Sun and the Earth, once every 90 minutes, what can it look at then?
Wikipedia lists several angles in https://en.wikipedia.org/wiki/Hubble_Space_Telescope
Observations cannot take place when the telescope passes through the South Atlantic Anomaly due to elevated radiation levels, and there are also sizable exclusion zones around the Sun (precluding observations of Mercury), Moon and Earth. The solar avoidance angle is about 50°, to keep sunlight from illuminating any part of the OTA. Earth and Moon avoidance keeps bright light out of the FGSs, and keeps scattered light from entering the instruments. ... There is a so-called continuous viewing zone (CVZ), at roughly 90° to the plane of Hubble's orbit, in which targets are not occulted for long periods.
So, +-50° away from sun (not ~180° as in Hohmannfan's answer) and some angle from Earth (and its limb) gives a lot of usable space for any moment of time...
Some older HST Primers have an exact value of prohibited angles (but they are not listed for cycle 24): ftp://ftp.stsci.edu/cdbs/perigee/jyounger/cp_primer/Ch_2_Systemoverview5.html#1876818 - Hubble Space Telescope Primer for Cycle 11, 2.4 Pointing Constraints
the telescope does not observe targets that are
- within 50 degrees of the Sun;
- within 15.5 degrees of any illuminated portion of the Earth;
- within 7.6 degrees of the dark limb of the Earth; or
- within 9 degrees of the Moon.
180 degrees of the sky is occupied by the Earth. 180 degrees of the field of view will cause sunlight to reflect in the optics of Hubble, causing unrecoverable damage. Nothing left then. The "downtime" is usually spent re-targeting and changing the observation setup. Even though non of the angles are as great as 180 degrees, the optics still requires a "shadow zone" margin of significant width, making the avoidance zone hemispherical in practise.
NOTE: This answer is actually wrong in three different ways.