If you use a bad engine or cheap fuel the rocket will run out of propellant long before it even reaches space (which is far away from the point you considered).

A key quantity of rocket motors is the specific impulse I_sp, which is the exhaust velocity divided by the (surface) gravitational acceleration. If a rocket has an I_sp of e.g. 500 seconds, then it needs to expel 1/500 of its mass as propellant to maintain its speed (close to Earth's surface). The mass flow decreases over time so we can assume ~1000 seconds of burn time for a fully fueled rocket that just maintains its velocity. If you want to get far away from Earth in 1000 seconds (let's say a 5 times the radius of Earth, ~30000 km) then you need a speed of 30 km/s. That's above the escape velocity.

500 s I_sp is better than any rocket engine that has ever flown, and it's at the edge of what chemical fuels could potentially do under ideal conditions. If you plug in more realistic I_sp values (~250-450 s depending on the propellant and use) then it gets even worse, and this is still using extremely optimized engines. A hobby rocket motor might have an I_sp somewhere in the range of 50-100 s.

tl;dr: Yes you could in principle reach space and even escape Earth with a slow rocket, but in practice no method has the absurd engine performance this would need.