My aunt has either "Medical Alert" or "Life Alert" - I'll find out which one, it's the one *without* the minimum required months in the contract (some of them have cancellation fees even if you end up hating the service so watch out). It is the old style "wear a box with a button that connects wirelessly to a device on your conventional phone line, it phones a monitoring center who you can talk to through the box". And yeah it's not cheap - it's at least $30 bucks a month.
If it was me, I'd carry a cell phone in a waterproof pouch around my neck (prepaid for emergencies works out to about $3 a month over 5 years - actually if you're just gonna dial 911 you don't even need a plan but I want to be able to call a tow or a cab or whatever), but if I was a lot older I could see that might not be so simple - she already has huge trouble seeing, dialing and hearing on a cell phone when she's OK, I wouldn't count on her being able to get it working in any reasonable time after a fall or stroke. For around the house my 2nd choice would have been a cordless handset, but I could not find an easily carryable cordless phone - they don't usually fold like cell phones so the buttons are exposed to accidental presses. I *could* put it in a hard shell box, and jury rig something to protect the keys, but the dang things are huge. The few small ones I found by searching have awful reviews and low battery time
There are "no plan" alternatives to the Life Alert type service - basically you buy the dialer/pendant, program in some phone numbers, and it will dial through those until somebody answers. The "Logicmark" one seems to get generally decent reviews (plus some frighteningly bad ones :p), but the family didn't look too hard at these 'cause nobody lives near enough plus a lot of us are Luddites who don't ALWAYS have our phones on so they went with the service plan.
EDITED: misremembered "Logicmark" even after JUST re-reading the reviews page :p
PS - whatever you get, test it ALL over the house/yard, walls/basements do funny things to wireless signals.