How many tacks fit in the plane?
First of all, the one-point compactification of three open intervals is not a "tack", it's a three-leaf clover. I think that you mean a one-point union of three closed intervals; of course it doesn't matter if the other three endpoints are there or not. This topological type can be called a "Y" or a "T" or a "simple triod". R.L. Moore published a solution to your question in 1928. The answer is no. It was generalized in 1944 by his student Gail Young: You can only have countably many $(n-1)$-dimensional tacks in $\mathbb{R}^n$ for any $n \ge 2$. For her theorem, the name "tack" makes rather more sense, but she calls it a "$T_n$-set".
Actually Moore's theorem applies to a more general kind of triod, in which three tips of the "Y" are connected to the center by "irreducible continua", rather than necessarily intervals.
I don't know whether I might be spoiling a good question, but here in any case is a solution to the original question (see as both Moore and Young did something more general that takes more discussion). Following domotorp's hint, there is a principle of accumulation onto a countable set of outcomes pigeonhole principle for uncountable sets. If $f:A \to B$ is a function from an uncountable set $A$ to a countable set $B$, then there is an uncountable inverse image $A' = f^{-1}(b)$. If you want to show that $A$ does not exist, then you might as well replace it with $A'$. Unlike the finite pigeonhole principle, which becomes more limited with each such replacement, $A'$ has the same cardinality as $A$, so you haven't lost anything. You are even free to apply the uncountable pigeonhole principle again.
Suppose that you have uncountably many simple triods in the plane. Given a simple triod, we can choose a circle $C$ with rational radius and rational center with the branch point of the triod on the inside and the three tips on the outside. Since there are only countably many such circles, there are uncountably many triods with the same circle $C$. We can trim the segments of each such triod so that they stop when they first touch $C$, to make a pie with three slices (a Mercedes-Benz symbol). Then, given such a triod, we can pick a rational point in each of three slices of the pie. Since there are only countably many such triples of points, there must be uncountably many triods with the same three points $p$, $q$, and $r$. In particular there are two such triods, and a suitable version of the Jordan curve theorem implies that they intersect.
The argument can be simplified to just pick a rational triangle that functions as the circle, and whose corners function as the three separated points. But I think that there is something to learn from the variations together, namely that the infinite pigeonhole principle gives you a lot of control. For instance, with hardly any creativity, you can assume that the triods are all large.
This is a well-known puzzle/problem, the trick is to make an injective mapping from any set of disjoint tacks to triples of $\mathbb Q^2$.
While the result of Young mentioned in the answer of Greg Kuperberg seems to be of purely topological interest, quite surprisingly, it has applications to study of the structure of the support of Sobolev functions:
T. Bagby, P. M. Gauthier, Note on the support of Sobolev functions. Canad. Math. Bull. 41 (1998), no. 3, 257–260. (MathSciNet review).
The paper of Bagby and Gauthier contains a proof of Young's result (Lemma 2 in the paper). The authors knew the result of Moore, but they were not aware of the generalization obtained by Young.