It is possible to make the switch, but there are a few components that probably won't be an even swap. This will depend on the vintage of the frame you're planning on using (and it really should be steel as aluminum doesn't age gracefully). Since I have a steel 1992 Bridgestone RB-T, I'll use it as an example.
The headset on your bike is most certainly 1 1/8" threadless, while the headset on an older bike will be 1" threaded. You will have to use a threaded fork and a quill stem. You may find it tricky to locate a quill stem to fit your handlebars if they have a larger clamp diameter (older bikes commonly use 25.4mm and 26.0mm bars).
Wheel size won't be an issue. 27" wheels have an ISO diameter of 630mm while 700c wheels have an ISO diameter of 622mm (4mm smaller at the rim). You'll want to make sure that your brake calipers reach the rim squarely, but that will depend on the frame. It really shouldn't be an issue unless you decide to run 650b wheels (which have many virtues).
A bigger problem may be fitting your rear wheel in the frame. Presumably your rear hub as an O.L.D. of 130mm (i.e. 130mm wide). Most older frames (such as mine) have 126mm dropout spacing to accomodate 6 or 7 speed freewheels. There are resources on re-spacing your dropouts by bending the stays (this is only possible on steel frame bikes.) Alternatively, you could buy a freewheel rear wheel, but then you'd have to futz with your indexed drivetrain and that's no fun.
You'll want to make sure that you use the bottom bracket length specified for your crankset which may require removing the old bb components from your new(old) frame. With the right tools it shouldn't be difficult.
Those few things will be your bigger obstacles. There may be one or two other small things, but they'll be easy to work around. If you're serious about this, also consider that there are a great many steel frames from the 80s that are worth the effort.