I would say that this is highly dangerous. It is against US and Canadian code to not have outside ventilation for any fuel-burning appliance in your home; that's your furnace, HWH and stove/oven, assuming all are NG or propane. It is only acceptable to have a "filter-only" vent hood for your stove if it's all-electric (which BTW is the case for every single apartment I've ever rented; gas appliances may be cheaper on utility bills in the US, but a gas stove is a huge fire hazard and general liability for any landlord).
The code is in place for a very good reason; not only can inefficient burning of fossil fuels produce carbon monoxide and smoke (both of which continue to cause damage long after you've reached fresh air), but even when these fuels burn ideally, they remove oxygen from the air and replace it with CO2. CO2 in itself is not toxic in the same way CO and soot smoke are; as soon as you reach fresh air the symptoms of CO2 asphyxia begin to dissipate, while soot and CO poisoning ("smoke inhalation") can kill you hours after you reach fresh air. However, the consumption of oxygen and production of CO2 in a space with inadequate ventilation is a double-whammy for anyone in the same space; the oxygen is being consumed so there's less of it even in upper strata of the room's air, and as the CO2 builds it settles downward in a "blanket", pushing oxygen up towards the ceiling and away from you.
As the CO2 level builds, your body's natural "inhale/exhale" reflexes go haywire in a Catch-22 condition called hypercapnia; your natural breathing while in a high-CO2 atmosphere actually increases the CO2 levels in your blood, but the only thing your body can do to reduce CO2 levels is breathe. So, you start hyperventilating, which only exacerbates the problem. Should you pass out from lack of oxygen, you will not wake up if someone else doesn't get you out of the room or get some ventilation of fresh air through it.
Your landlord is running illegal housing. However, he may not know it, so be nice at first. Follow standard procedure for maintenance requests, and ask the landlord to install a proper outside vent line for this fume hood. If he refuses or drags his feet, you can call in the city's Code Compliance officials, or federal HUD (Housing and Urban Development) representative, and they will MAKE the landlord comply. Depending on the terms of your contract, the landlord may be giving you a free out by not living up to his end, meaning you may be able to break the lease at no cost if this has gone on for some time with the landlord's knowledge and inaction.
Understand that the cheapest way for your landlord to fix the problem with tenants still occupying the units may well be to cap off the gas feed and replace all the gas cooktops with the cheapest electric setups he can find. If this was the reason you moved in, and you don't get a "free out" from this debacle, you may find yourself stuck with a spiral-coil POS.
Any house can be equipted to run a gas stove. Some houses have gas service from the utility, others have on-site tanks.
If you already have gas service or some type of gas distribution in your house, then you can add a stove by having a licensed contractor run a line to your kitchen. The process is similar to running new pipes or electrical circuits. It is easier to do if the area under your kitchen is not finished, otherwise expect a lot of holes. The cost will depend on the distance from the gas main and the complexity of the run.
Cost will also depend on where you live. Just for an example, in Toronto, ON, Canada, it cost me $700 CAD to add hook-ups for my BBQ and stove. The BBQ run was outside while the stove run was done in my unfinished basement.
Make sure you contact a licensed installer/contractor. Don't mess with gas yourself.
Best Answer
In rural locations in the US, propane is often used as a fuel for cooking and heating. The gas is stored in an outdoor tank and piped into the house. Propane burns hotter than methane ("natural gas"). Therefore you may need a special stove to use propane, and if you want to hook up appliances such as a water heater or clothes dryer, you will definitely need models designed for propane.
Piping for gas is a high-stakes job: if you mess it up the odds of it killing or hurting you are significant. Your local law almost certainly requires that only licensed professionals work on gas lines for this reason. With propane, there may also be special ventilation or safety system requirements. So for all these reasons you should start by calling some vendors/installers and asking for proposals from them. You want to figure out whatever is common in your area, and do that -- sticking with whatever's common will make it easier to shop around for fuel deliveries and repairs, and doing something unusual or amateur around gas lines is a sure way to scare off potential buyers if you ever sell your house. In fact, if your system wasn't installed professionally and passed government inspection, I'd be surprised if anyone would even sell you fuel.
So although this is the DIY stackexchange, some jobs are really only for professionals. Get recommendations from neighbors and call around.