What will happen if I leave out hardboiled eggs in water for four weeks?
If you boiled the eggs but left them in the pot of water, you may be in luck. The shell should keep the rotten egg safe from cracking and exposing the apartment to the truly rancid smell of spoilt eggs, assuming they haven't cracked during boiling.  


  You will know a spoiled egg when you crack open the shell. Whether raw or cooked, a spoiled egg will have an unpleasant odor. No question about it: throw it out.
  - Net Wellness


What you will probably find when you get home are unmolested eggs in a stagnant pot of water.  Throw the eggs out if they haven't cracked.  They WILL be spoilt. This answer describes why.


  According to the USDA (similar information can be found from other food agencies):
  
  
    Why do hard-cooked eggs spoil faster than fresh eggs?
    
    When shell eggs are hard cooked, the protective coating is washed away, leaving bare the pores in the shell for bacteria to enter and contaminate it. Hard-cooked eggs should be refrigerated within 2 hours of cooking and used within a week.
  
  
  Simply put, you're damaging the shell, and that expedites spoilage.


If you're unlucky and the eggs did crack...well, the water might absorb some of the scent, but after four weeks, you're still likely going to have some odor.  Throw the eggs out, pour the water out, throughly clean the pot, and try to air your apartment out as best you can.