General synonyms:
* Space
* Separation
* Gap
* Span
* Interval
* Reach
* Extent
* Range
* Spread
Synonyms emphasizing physical distance:
* Length
* Width
* Height
* Breadth
* Depth
Synonyms emphasizing emotional or social distance:
* Alienation
* Estrangement
* Disconnection
* Isolation
* Removal
* Detachment
Synonyms emphasizing temporal distance:
* Duration
* Period
* Interval
* Span
* Lapse
* Length
Synonyms for "keeping a distance":
* Avoiding
* Staying away
* Steering clear
* Holding back
* Withdrawing
The best synonym for "distance" will depend on the specific context of your sentence.