We aim to keep 6 months of expenses. The rationale is that its enough time to recover from most serious illnesses (that you can recover from) or a redundancy or pay for a large unexpected problem not covered by the insurance (e.g. the boiler dying).
It also gives us enough time to reorganise finances if needed. For example we could get out of contracts (like mobile phones, sky TV), sell the car, and maybe even find a cheaper house if needed in that time.

It will take a good chunk of time to build up that amount and it's worth considering how many commitments you have (kids, wife, mortgage, car...) as the fewer you have the less you need.
If you have fewer commitments you can be comfortable with much less contingency. When I lived in rented accomodation and didn't run a car or have many possessions, I just maintained enough cash to cover my bills for about 6 weeks, this would give me enough time to find another job, and if I didn't get one I could always crash round a friend's house.

I think this varies considerably depending on your situation. I've heard people say 6 month's living expenses, and I know Suze Orman recommended bumping that to 8 months in our current economy.

My husband and I have no children, lots of student loan debts, but we pay off our credit cards in full each month and are working to save up for a house. We've talked through a few different what-if scenarios. If one of us were to lose our job, we have savings to cover the difference between our reduced income and paying the bills for 6 or 8 months while the other person regained employment. If both of us were to lose our jobs simultaneously, our savings wouldn't hold us over for more than 3 or 4 months, but if that were to happen, we would likely take advantage of the opportunity to relocate closer to our families, and possibly even move in to my parent's house for a short time. With no children and no mortgage, our commitments are few, so I don't feel the need to have a very large emergency cash fund, especially with student loans to pay off.

Think through a few scenarios for your life and see what you would need. Take into consideration expenses to break a rental lease, cell phone contract, or other commitments. Then, start saving toward your goal.

Also see answers to a similar question here.

How to start is pretty simple. With your next pay check set aside an amount and open a separate savings account. Since this is an emergency fund - you want it someplace where you can get to the money quickly (so a CD or mutual fund is not good), but you want it in a separate account so that you don't accidentally use it.

Once the account is opened I'd recommend setting up an automatic transfer, or make it part of the direct deposit if you do that, so that you put in some money regularly (every pay check). By adding to it regularly and not using it, you'll more quickly achieve your goal.

I'd recommend stopping, or slowing any retirement savings or other investing, until you get the emergency fund in place. If you have an emergency, the money in the retirement fund isn't going to do you much good as it costs too much to do an early withdrawal. The whole point of the emergency fund is to have liquidity when you need it so that you don't incur the costs of unplugging your longer term investments.

Also don't worry overly much about making money on this money. This isn't an investment it is there for emergencies.

6 to 9 months worth of expenses is recommended. You should also consider having long-term disability insurance in place, in case of serious illness or accident.