Credit score isn't solely tied to credit cards.
I'm with Rammy - emergency money is in savings. Most logo-ed debit cards can be used for hotels, rentals, etc.
The only "credit" I run right now is a vehicle loan, which I have directly withdrawn the same day I have direct deposit. No stress, no thinking, it's paid, and for fools like me who can't keep track of anything, I know it's paid on time, all the time. Just doing that is raising my credit score quietly and easily. I could pay it off right now, but I would like to raise my score from the mess my ex left me. Unfortunately many places do depend on your score - car insurance etc - but once that's set up, then I'll just go back to paying bills and letting that show my payment history. I've got some time before I can get everything paid up (shelling out about $25,000+ instead of bankruptcy), but I should be quite solvent at the end, with good history. Goal is to pay off that amount in about 2-3 years or less, which I can do now. I couldn't do it before, which now leaves me with a court order and garnishments and crap (still researching if I can sue my ex). Once I finally made enough I could actually afford to start paying off the debt, they released all of the garnishments and I have an automatic payment sent. If one is dealing with horrible debt issues and whatnot, once you have income, it's a lot easier to call up and say, "I'll give you X a month, and Y down right now." and call you own plan and amount.
I had close to 800 credit when I met my ex, with no credit card, just a series of car loans and paid bills. Didn't listen to daddy's best advice - "Always get a complete financial report before marrying someone." Left the marriage in a year with a score in the mid 500s, and I've been in there since. Good job now, and things are changing, so goal to get in the 700s again by this time next year. With no credit cards.