The amount needed depends on how much you make, how much you need to live on and other factors.. Yes, you should get disability insurance. First place to look is what is offered through your employer. Take advantage of that and elect the maximum coverage they offer. Find out whether that is a pre or post tax deduction. Then work with a professional to find the "gap" in your income which will dictate how much you need for your own personal policy.
The best way to think about this is that you’re trying to replace your net income in the unfortunate case in which you become disabled.
There are two types of disability policies. Short term that covers you up to 6 months, and a long term one that can be until you reach full retirement age
Great answers from everybody. Disability insurance is very important to have in place but it is not discussed much. Life insurance is much more common, but you are much more likely to become disabled causing a loss of income which will severely impact your life. Check your employer or any affiliations/groups you may be associated with as getting this insurance through one of these avenues is much cheaper than going out and getting a personal policy on your own.
Everyone has given great answers thus far but there is just one more thing I would like to add. You also have to decide if you want to insure for "full disability" or "full and partial disability." If you are looking to insure against the worst case scenario, you go with full only, which covers catastrophic events that may cause you to never work again. This will keep the coverage relatively low cost.
However, you have to realize that the vast majority (something in the range of 60-70%) of all disabilities are partial disabilities -- meaning you are out of work for an extended period of time but will eventually go back on either a full-time or at least partial basis. Choosing to insure for this will cost more than "full only" but offer more protection for a wide variety of disabling scenarios.