Could You Keep Accelerating Forever?

yes. you can accelerate forever. your rate of increase in absolute speed will simply dimish as you get closer and closer but never actually reaching the speed of light.

Is constant acceleration possible?

Feasibility. A major limiting factor for constant acceleration drives is having enough fuel. Constant acceleration won't be feasible unless the specific impulse for fuel (the fuel's fuel efficiency) becomes much higher.

Is there a limit to acceleration in space?

Nothing can travel faster than 300,000 kilometers per second (186,000 miles per second). Only massless particles, including photons, which make up light, can travel at that speed. It's impossible to accelerate any material object up to the speed of light because it would take an infinite amount of energy to do so.