A well-known truism in programming is 'Don't use Exceptions for flow control'. Your code should be aware of the restrictions and guard against them before calling the constructor rather than handling errors. Exceptions exist for expectational circumstances, especially ones that cannot be predicted or guarded against (for example, an IO stream may become invalid during writing, despite being OK during a previous check).
While you can throw exceptions in your constructor, this is not always ideal. If you are writing public objects that you expect to be used/reused by others, Exceptions are the only real option for public constructors, however such limitations and their result (e.g. what exception will be thrown) should be clearly documented in the javadoc for the class.
For internal classes, assertions are more appropriate. As Oracle states: "Assertions... should be used to check for cases that should never happen, check assumptions about data structures, or enforcing constraints on arguments of private methods."—Using Assertions in Java Technology. You probably should still document your expectations for the class, but your application should internally do any checks beforehand rather than relying on any Exceptions being thrown.
Static factory methods can help a little, their benefits are elaborating upon a bit by another question: How to use “Static factory methods” instead of constructors. However, they don't give strong validation options without, again, relying on Exceptions when things are not valid (that, or returning null, which is less informative).
Your ideal solution is the Builder pattern. Not only does it allow for a greater deal of flexibility in managing your arguments, you may validate each one individually, or have a validate method that can evaluate all the fields at once. A builder can and should be used to hide the object's actual constructor, enjoying sole access to it and preventing any unwanted values from ever being submitted, while assertions can guard against 'the builder should never submit these values'.