A fifty-year-old house isn't really that old, and a lot depends on how the home was cared for during that time. After all, you can find fully original (or restored) classic 50-year old cars that need nothing to drive as well as total rust buckets that should have been off to the junkyard a long while ago.
That said, a lot has changed since the Rolling Stones sang "Ruby Tuesday," and a completely original 50-year-old home could have a number of items that need attention.
Energy efficiency was rather different then than today. Insulation materials have changed and improved over the years, but unless someone opened up the walls at some point, you probably have (by today's standards) sub-par insulation in the home.
Windows back then were less efficient in many ways; today's modern double-pane thermal windows are self insulating, removing the need for triple-track storm windows. That said, they are more fickle in some ways and they can breach and cloud, but at least you never have to reglaze them with glazing putty and linseed oil.
Of course, heating and air conditioning efficiency has hugely improved, too. New systems no longer even require traditional chimneys, as they capture and employ virtually all the heat produced by whatever process is in the home. In addition, asbestos fiber was often used as insulation on older heating systems and may be a concern; in general, however, these should usually be in a stable state and won't present a hazard unless disturbed. Certainly, old air conditioning units (wall, window or central) have all been superseded by higher efficiency units.
A 50-year old home is more likely to have 2x4 construction framing on 16" centers and 2x8 or 2x10 roof beams. It is also more likely to have plywood rather than OSB or other sheathing, and may even have solid wood tongue-in-groove sheathing. These construction forms can be more resilient to getting wet and such and are durable.
Plumbing systems have evolved, too. Gone are 3 and 5 gallon per minute showerheads; new standards save water. This is also true of sink taps and especially toilets, where 5 gallon flushes have given way to 1.5 gallon in efforts to save water. Galvanized and cast-iron waste pipes are now usually plastic of some form, as are vent lines, and copper and lead-soldered waterlines have given way to PEX or other plastic systems.
Your electric service may be insufficient to meet today's lifestyle. Older homes may only have 100 amp service; 200 amp or more is common today. A 50-year old home will likely have circuit breakers rather than old-style fuses but still may have only a limited number of circuits, meaning more chances to inconveniently overload (trip) a breaker when plugging stuff in that draws a load -- like a hairdryer in the bathroom or toaster in the kitchen.
Paints have different composition today than back then, as well. Oil-based enamels were far more commonplace then than now, when almost everything is latex-based. It's important to know that lead-based point wasn't banned until 1978, so a 50-year old home may (or may not) have both interior and exterior surfaces that can require special attention. Also, painting over oil-based paint (lead or not) with latex requires some special preparation and priming.
Home design is different, too. Older ranch-style homes and cape-cods gave way to more bi-levels as the suburban migration continued. There is greater emphasis given to closet space and bathroom size today, for example, and modern floorplans can tend toward more of an "open" layout than not.
There can be other items, too. The reality is that there may be more or fewer "pitfalls," but a lot depends upon maintenance, upkeep and upgrades that were performed on the property over time. You don't need to rip open the walls and re-insulate, or tear out the plumbing and modernize it all. There can be benefits to doing so and even savings to be had over the long haul for having done so (plus up-front cost and inconvenience, too) but you can live with inefficient windows and subpar insulation -- as did everyone who owned the house before you.
Depending on what's been done before, you may have a lot to do (or virtually nothing) to have a long-term, trouble-free ownership period. The age your home was built is relevant, as this is the age of the foundation and overall structure, but the age of major components and systems is probably a better gauge of the "age" of the house and a better guide to any pitfalls or issues you may encounter.