Excellent post by Loopy.
Let me share a story, my friend was a doula recently for a home birth. She's a very experienced hospital doula (in the US) but this was her first home birth and by the sounds of it, tricky. The baby came in an awkward presentation (I think perhaps, shoulder dystocia) and the midwife handled it excellently, guiding the baby down. Then the woman started to haemmorrhage and the midwife dealed with that calmly and efficiently too. My friend who has worked for the hospital for many years said she'd never seen anything like it. At both points in the hospital, all hell would have broken loose and likely a caesarean would have happened.
She said witnessing the home birth and the midwife's work was life changing and may seriously change how she works.
Now the contrast between hospital and home isn't quite so massive in the UK, but I think it's worth bearing in mind that hospitals can both cause problems and make bad situations worse. Midwives who are giving you one-on-one care are likely to spot and resolve problems earlier and they carry resuss equipment for you and the baby, as well as drugs for haemmorhages etc.
Loopy is so right that there are risks in both locations. No birth is 100% safe. But for me, I would (and did) opt for home birth. Not for any hippy ideals but because I judged it to be the safest place for me.
Remember there have been very prestigious studies that consistently find home birth to have the same mortality rates as hospital birth and considerably lower morbidity rates.