My house rule for death saving throws can be useful for 5e or other "modern" editions, but is way too merciful for old school, gritty games, where 0 HP usually carry very severe consequences. For this kinds of games, you can use something a tad more extreme. There are a couple of options below; you can use them in combination with the original post where applicable (specially, any rules for immediate death upon -10 HP, etc, still apply).
Option 1) If you reach 0 HP, save vs death. If you fail, you die. If you succeed, roll 1d6.
Option 2) If you reach 0 HP, roll 1d6.
1-2. Dying. Roll again every turn.
3-4. Disabled. Roll again if you try any significant effort (moving more than half speed, fighting, etc).
5-6. Unconscious. You can be awakened with ease by other characters, but if you aren't roll again after 10 minutes.
In any case, all dice rolled are kept on the table until you can get decent rest and healing (which usually means recovering HP). Rolling the same number twice means death.
Alternatively, you can use the table below, that includes more varied results.
5. Lasting injures to limbs, organs or abilities*.
6. Roll again, with 6s counting as "nothing happens for now".
* Roll 2d6 to find how many weeks until it naturally heals. If you roll the same number on both dice, it is permanent.