My take on travel-related emergencies based on years traveling & living overseas while in the military and doing some international athletic competitions totally solo where I couldn't rely on military medical/transportation backup. I've also spent month s on the road in a schoolbus/van touring the States after I retired.... so take the following for what it's worth to you. YMMV
1. Family at home issue - you need to get home YESTERDAY for whatever reason. Find a LBS(hotel, local, whatever) and arrange to leave bike there/store it. Worry about getting it home later. It's that simple. First thing's first.
Be prepared to write off the bike if it comes to that. Sometimes it might cost more to get it back home than it's worth.(I wrote off $650 skis under similar circumstances once....back in the late 70s/early 80s, I forget exactly... yes, I was a serious racer in a past life.)
You need to have the cash and/or access to credit and/or your bank account to buy a "fly today/tomorrow" ticket - usually much more expensive than the buy_now-travel_in_6_months type of ticket. Sucks but you just do it. Happened to me 2x in 14 years.
[Having an open-return date ticket never seemed to work when I had to return quickly/unexpectedly.]
2. Injury - you need medical insurance that covers you for the area where you are traveling. No insurance (or it doesn't cover you/is refused in a particular place)? This is where a wad of traveller's checks/$100 US greenbacks comes in (yeah, that can be a safety/theft issue.. but that's another story). You pay as much as possible in the traveller's checks first and the cash second.
In the States, it's not as big a deal as overseas, but I was once refused medical care at an Austrian clinic and told to head down the road to a hospital emergency room. In the States, I've never been turned away from an emergency room even when my medical coverage was iffy. Otoh, the check-in staff can be real dicks. But, if you are bleeding, they wil definitely take care of you first and then bleed your wallet dry later.
3. Bike problems - assume (near)catastrophic failure of some part of the bike. Buy/carry a replacement te entire length of your tour. Get to a train/bus station and go from there to nearest town/city where a replacement can be found/made. Hit up a local welder to have your frame/rack/derailleur hanger re-welded.
Basically, you need to decide what are the potential showstoppers for which you are willing to carry spares and the likelihood that you can buy replacement parts where you will be riding. If you're in the hinterlands, parts may be unavailable. Can you afford (money/time-wise) to just sit and wait while someone ships you a replacement (from home/from online)? I've carried far more than others on various routes at times - but I've never regretted it for longer than an individual hill. I'm the conservative type - I 'd rather have it and not need it than not have it and be stranded/tour-ended.
4. Illness - relates to but is different than injury. Sometimes you can just go to a local clinic and get help cheaply. Other times, you're going to face a serious decision - hole up in a hotel/motel and be sick or (try to) return home while ill. The latter is something that can be a gamble - I've been refused boarding when I was sick and trying to fly out from Istanbul to Frankfurt.
Sometimes, Montezuma's Revenge is just that - a couple day's discomfort and you can move on. Other times, it's something else and only a medic can tell. What's your personal medical knowledge? Mine's pretty good but I know its limitations.
If you're overseas, what's your medical language ability? Can you convey your symptoms to the medical types? I always carried a local language medical reference card or pocket dictionary. At one point, my medical-travel library was quite extensive.
Last thing about medical insurance. In the event of catastrophic injury, does your insurance cover medical flights home? Some policies don't. Medical flight expenses can be unbelievably expensive.
All that said, I have never let any of those concerns stop me from traveling.