I have a problem with my house such that with all my windows and doors closed the house smells and the air feels stale. I'm thinking of purchasing a high efficiency HVAC to vent the stale air out and blow fresh air in. Would a HVAC accomplish what I'm asking? Or is the only way to vent stale air out is to open the windows?
HVAC systems don't exchange air with the outside unless you specifically add an air exchanger. In older houses, this wasn't necessary because they leaked enough that you always got new fresh air coming in somewhere. With newer houses and recently fully renovated ones, the houses can indeed be sealed up so air tight that an air exchanger is necessary.
Based on your response, what it sounds like you have is an old house, recently renovated to be made air tight, with existing duct work and a furnace.
Based on that, you don't need a new HVAC system, you need an add on air-exchanger, such as this:
Ducting will need to be added to the outside to allow this machine to suck in outside air and blow out inside air. This kind of device is, IMO, best installed connected to air return of your existing furnace so that fresh air can be injected, heated, and distributed thru the house.
HVAC may make a difference, especially if you buy an air filtration system with it, but I wouldn't buy one on that basis. You buy an HVAC if your house is too warm or too cold, not if the air is stale. For that I'd do a good cleaning, shampoo carpets, fix any damp problems, and buy a couple of portable HEPA air filters. Probably more effective, and certainly much less expensive than buying an HVAC!