Does it mean that absolutely nobody can land there until conditions improve? Can ATC still give you an IFR clearance to your destination airport in this case?
It would be very helpful to know where you read or heard the phrase, to get some context. But, it likely means that weather conditions at the airport are below the IFR takeoff minimums in 14 CFR 91.175:
(f) Civil airport takeoff minimums. This paragraph applies to persons operating an aircraft under part 121, 125, 129, or 135 of this chapter.
(1) Unless otherwise authorized by the FAA, no pilot may takeoff from a civil airport under IFR unless the weather conditions at time of takeoff are at or above the weather minimums for IFR takeoff prescribed for that airport under part 97 of this chapter.
(2) If takeoff weather minimums are not prescribed under part 97 of this chapter for a particular airport, the following weather minimums apply to takeoffs under IFR:
(i) For aircraft, other than helicopters, having two engines or less—1 statute mile visibility.
(ii) For aircraft having more than two engines— 1⁄2 statute mile visibility.
(iii) For helicopters— 1⁄2 statute mile visibility.
Landing minimums are per-approach so it wouldn't make sense to say that the whole airport is below IFR minimums based on landing criteria. However, without reading or hearing the phrase in context, I may be wrong.
As for landing there anyway, even in zero visibility a private pilot operating under part 91 is free to take off, or to start an instrument approach. You can't legally land if visibility is below the approach requirements, but you can still fly the approach to 'take a look'. It might not always be smart, but it's legal.
On the other hand, commercial flights under part 135 or 121 can't even start an approach unless conditions are above certain minimums (e.g. see 135.225(a)). Those minimums are usually just the ones required for the particular approach, but they can also have operator-specific requirements. Of course, many commercial aircraft have crew and technical capabilities that light aircraft don't (like shooting a CAT III ILS), so they might still have a better chance of landing in marginal conditions.
I think they are referring to the minimum visibility and ceiling that is required. There are two cases. Actually landing at an airport and filing an IFR flight plan.
Per §91.175 Takeoff and landing under IFR.
(c) Operation below DA/DH or MDA. Except as provided in §91.176 of this chapter, where a DA/DH or MDA is applicable, no pilot may operate an aircraft, except a military aircraft of the United States, below the authorized MDA or continue an approach below the authorized DA/DH unless—
(2) The flight visibility is not less than the visibility prescribed in the standard instrument approach being used; and
Part 91 aircraft may attempt an approach if visibility is reported to be below minimums, but Part 121 and 135 may not begin the approach if reported visibility is below minimums. Effectively, the airport is closed to them.
You may file to an airport that is expected to be below minimums at the time of arrival, but you must file an alternate that is above IFR minimums. §91.169 IFR flight plan: Information required. has details.
(c) IFR alternate airport weather minima. Unless otherwise authorized by the Administrator, no person may include an alternate airport in an IFR flight plan unless appropriate weather reports or weather forecasts, or a combination of them, indicate that, at the estimated time of arrival at the alternate airport, the ceiling and visibility at that airport will be at or above the following weather minima:
(A) For a precision approach procedure. Ceiling 600 feet and visibility 2 statute miles.
(B) For a nonprecision approach procedure. Ceiling 800 feet and visibility 2 statute miles.
So yes, ATC can give you a clearance to an airport that is below IFR minimums.
You can still get clearance to the destination, at your discretion. It's not unheard of for weather conditions to be below minimums now and quite acceptable a few hours later after all.
Think Milan Linate, an airport plagued by early morning fog. A flight taking off from say Amsterdam to Milan may take off with the airport covered in dense fog that would make it impossible to land legally, and by the time it gets there it's a clear morning with no fog or clouds.
Of course as always you, the pilot, are responsible for your actions and if you take off knowing your destination is currently closed for you you'd better have a viable alternate plotted just in case the expected improvement in conditions doesn't happen.