Is it legal to land in bad visibility conditions without ILS?
The answer is "Maybe".

An ILS is actually comprised of two signals - a localizer (which lines you up with the runway) and a glideslope (which guides your vertical descent).
If the glideslope is out of service (as it was when KAL801 was flying its approach) you fly a localizer-only approach, which requires visual contact with the runway at a higher altitude.
In this case as shown on the plate below the ILS can be flown down to 531 feet (in red); the Localizer-Only approach can only be flown down to 1440 feet (in green).



So as long as the crew had the minimum visibility and cloud height requirements (and observed the altitude restrictions on the approach) they could legally fly into the airport on a localizer-only approach



In the case of KAL801 there were a few contributing factors to the accident (go ahead and skip to page 142, unless you really like reading NTSB reports):


The glideslope may have been receiving false signals.
The crew should have disregarded the glideslope indication anyway as it was NOTAM'd out of service, but it's possible the crew followed the needle anyway.
The DME fix for the approach is not located at the airport.
This may have caused the crew to believe they were closer to the airport than they actually were.
Because of (1), (2), or some other reason, the crew flew below the Localizer-Only minimum altitude.
This is ultimately why they hit the hill -- they were too low during the approach.
Had they been at or above the localizer-only minimum altitude they would have cleared the obstacles.