So let’s get this straight. Aliens so advanced that they can construct interstellar craft one quarter the size of our moon, yet so obtuse they are unaware of the consequences to themselves as well as us of approaching the earth or ANY other planet in a craft of such size. So the ships are in effect empty bubbles of no appreciable mass (much like the head that would devise a scenario so ridiculous). Doesn’t it embarrass you to print up nonsense like this?

Size or mass? It makes a difference.

They would probably be mostly hollow, to make room for, well, rooms and hallways, and cargobays and stuff.
Likely also be made of very light materials, and I will also assume some form of anti-gravity technology to reduce strain on the structure, and energy requirements when accelerating.
I therefore conclude that we they could come as close as they want.

The ships could stay the same distance that the moon is: About 240,000 miles.

With a invisible shield they could approach any world easily.

@kritiper is that science, or just a random answer you felt like giving just because you can?

Would the shield eliminate the mass? The question itself is so maddeningly naive and missing in essentials as to be a waste of imagination.

I like this Q because you are really making me think. I have to agree with @ragingloli ‘s assumption that the craft would be much lighter than it looks. If it were ¼ the size of the Moon and had the same density then it’s mass would be ¼^3 = 1/64 the Moon’s. But it likely is much less dense.
Let’s do a sense check and calculate density of something we know. How about my Subaru Forester. Length 4.6m, Width 1.8m, height 1.7m. Volume if a simple cube is 4.6×1.8×1.7=14.1 m^{3 Since it really is not a cube let’s knock off a bit and make it 10 m}3.
Mass is 1522 kg, so density is 152 kg/m3.
Let’s compare that with Earth and the Moon. The Moon’s density is ~3300kg/m3 or 3.3 g/cm3 The Earth’s density is ~5500kg/ m3 or 5.5g/cm 3. So a flying Subaru would have a density of ½0 of that of a similar sized chunk of the Moon. Since it is 1/64 the Moon size its mass must therefore be 1/64×½0 or 1/1200 of the Moon. But alien technology is way more advanced than a Subaru and it does not have to pass crash testing. So I will just estimate they are 1/10 of the density of a Subaru.
That brings the mass of their craft to 1/12000 of the Moon.
Now gravity is a function of the square of the distance. Rounding numbers here the Moon is 200,000 miles away. If they were in low earth orbit the closest they could be is 4000 miles or 1/50 of the distance. Squaring that you get 2500 times the force if they were at the surface of the Moon. So, their craft is 1/12000 mass of Moon but proximity exerts a force 2500 times so the net is 2500/12000 = 0.21 the Moon’s pull.
Expect tides to be 21% higher if they were in the right place at the right time.

~Note: All number are +/- 0.786%

@LuckyGuy so, 12 of them over one quadrant of Earth would be enough to create an enormous tidal wave with just their presence.

@Ltryptophan It is science that the moon is about 240,000 miles from Earth. I thought everyone knew that…

@kritiper Are you concluding that the closest they can come is the same distance as the moon?

If yes, how are you making this conclusion? Thanks

@Ltryptophan I went with your assumption that they are ¼ the diameter of the Moon.
Again for easy math let’s say the Moon is 4000 miles in diameter and Earth is 8000. The closest they could get to the surface would be 500 miles. There is not enough physical space to put all of them together in one spot. Maybe they can string themselves out in a line between Earth and the Moon to really do damage.

@Ltryptophan I’ve been giving this a little more thought. The force of gravity is proportional to the product of the masses and inversely proportional to the square of the distance. That means the crafts would be subjected to the force of Earth’s gravity as well as their own as they got close. Their levitation capability and structural members would need to support and resist these tremendous forces.
Maybe they have some kind of an anti-gravity shield to block Earth’s gravity. If that is the case their own gravity would not pass through the shield.
Unless….. it was a one way shield! Hey! They could use that as a form of propulsion! They aim the shield at a large mass, like a star and let its gravitational force pull them while blocking the gravitational forces from everywhere else.
Let’s work on that project. Maybe PPG has a paint that would work. ;-)

We would have slightly moist summer mornings.

@Ltryptophan No. If they were concerned as to what effects their craft may have on Earth, the distance to the moon might be a good safety factor. Plus, they wouldn’t know what danger their proximity to us would mean for them, so they might want to stand off a ways in case we launch some kind of defensive/reactive response.

First read of the question, I picture @LuckyGuy dashing to retrieve compass, protractor, and calculator.
I see his answer, and I read the whole thing, knowing not one bit of it will mean anything to me. I have a picture of this attractive, middle aged geek, gently biting a knuckle between calculations.

Then came his second post, with his own version of string theory. I’m already intimidated by his opening sentence of the first post. I know I’m not going to attempt an answer, if I actually made a serious attempt it would just look silly after all that math.

I now have a picture of him doing a happy dance which seems oddly familiar, and I start humming. iiiif I were a rich man, Lidle didle didle dum…
Then I get to a third post. He has partially exposed what I thought was a new idea, which I used in one of my books.
Now I don’t know if I should have a hissy fit, and scrap the whole project, or gloat because @LG thinks it could be feasible.

Anyway, if I was going to bring a fleet of ships to an unsuspecting planet, I would coordinate my arrival with a comet. We could angle out in a direction which uses the comet and tail for cover. Boom! Seemingly sudden appearance. It works great whether I want to keel you, but also if I want a favor so I’m aiming to impress. In this scenario, distance doesn’t matter.

“attractive, middle aged geek, gently biting a knuckle between calculations.”

I am not offended at all. :-)

But this question SHOULD read, how close can the spheres get before they are torn to pieces? And for this, there are so many missing parameters that the question is silly in the extreme. You have to invent supposes that rival fairy tales.

From the responses I’ve received from the OP, this is a very serious question. Except for the “alien” part…