There is actually nothing weird about this at all. What you have discovered is that **for continuous quantities, information is scale-relative**. Or, to put it another way, the "true" entropy of *any* continuous quantity not known *exactly* (or, known down to a finite number of *exact* candidates) is $\infty$. Which, of course, should make intuitive sense to you - it takes an infinite number of digits to name an arbitrary real number, and there are an infinite number of potential candidates (uncountably infinitely many actually, $\beth_1$), whenever you have a whole continuous smear of a probability distribution that is a nontrivial continuous function.

But, of course, just assigning "$\infty$" all the time would not be very useful. Hence, we have to introduce some form of relativity, by which we measure the information versus a moveable reference level. It's kind of like decibel measurements - $0\ \mathrm{dB}$ as an absolute measure must always be set arbitrarily for similar reasons.

The simple definition of entropy you give, i.e.

$$H[X] = -\int_S f_X(x) \lg f_X(x)\ dx$$

does this in a very simple manner, which can be "reverse engineered" from it by feeding it a suitably-wide uniform distribution: namely if you feed it

$$f_X(x) = \begin{cases}\frac{1}{b - a},\ \text{if $x \in [a, b]$}\\0,\ \text{otherwise}\end{cases}$$

you will find that it gives you $\lg(b - a)$ and this is **1 bit** exactly when $b - a$ is **2**, in other words, when you have a situation in which the position is *known to within two units*. When it is two bits, it is *unknown to four units* and, in general, $n$ bits, unknown to $2^n$ units.

That is, we have a uniform grid, formed by whatever measuring unit we are measuring $x$ in (e.g. meters, centimeters, millimeters, whatever), and the entropy is how much worse you are informed compared to knowing "to the nearest unit", as in a simple "digital" or "quantized" sense. Hence, it is no surprise you can have a negative entropy if you know the position *better* than the nearest unit! E.g. if your base unit is meters and you know it to $\frac{1}{128}$ meter, then you should have an entropy of -7 bits and, indeed, you do! But the entropy can keep getting more negative, which is what I was saying before, because there's still infinitely more information (at least in theory) you *could* still have yet to know about where exactly that thing is (or isn't) located!

And if you want to make this more explicit, if you have a set reference scale you want that is different from your measuring unit, i.e. you want "entropy relative to 0.001 m" when your positions are measured in m, you just add that to $H[X]$:

$$H_\mathrm{adjust}[X] = -\int_S f_X(x) \lg f_X(x)\ dx - H_0$$

where $H_0 = \lg(l_\mathrm{ref})$, and $l_\mathrm{ref}$ is your reference length measured in your unit of choice. So if your measured in meters that you know to 2 m, i.e. $H[X]$ is 1 bit, then relative to 0.001 m, where that $H_0 \approx -9.96$ then $H_\mathrm{adjust}[X] \approx 10.96\ \mathrm{bit}$.