Q-learning for continuous state spaces
Yes, this is possible, provided you use some mechanism of approximation. One approach is to discretise the state space, and it doesn't have to be to a small value. Provided you can sample and update enough times, then a few million states is not a major problem.
However, with large state spaces it is more common to use some form of function approximation for the action value. This is often noted $\hat{q}(s,a,\theta)$ to show that it is both an estimate and that you are learning some function parameters $\theta$. There are broadly two popular approaches to Q-learning using function approximation:
Linear function approximation over a processed version of the state into features. A lot of variations to generate features have been proposed and tested, including Fourier series, tile coding, radial basic functions. The advantage of these methods are that they are simple, and more robust than non-linear function approximations.
Neural network function approximation. This is essentially what Deep Q Networks (DQN) are. Provided you have a Markov state description, you scale it to work sensibly with neural networks, and you follow other DQN best practices (experience replay table, slow changing target network) this can work well.
Q-learning for continuous action spaces
Unless you discretise the action space, then this becomes very unwieldy.
The problem is that, given $s,a,r,s'$, Q-learning needs to evaluate the TD target:
$$Q_{target}(s,a) = r + \gamma \text{max}_{a'} \hat{q}(s',a',\theta)$$
The process for evaluating the maximum becomes less efficient and less accurate the larger the space that it needs to check.
For somewhat large action spaces, using double Q-learning can help (with two estimates of Q) - this helps avoid maximisation bias where picking an action because it has the highest value and then using that highest value in calculations leads to over-estimating value.
For very large or continuous spaces, it is not usually practical to check all values.
The alternative to Q-learning in this case is to use a policy gradient method such as Actor-Critic which can cope with very large or continuous action spaces.