Can a constructor in Java be private?

Yes, a constructor can be private. There are different uses of this. One such use is for the singleton design anti-pattern, which I would advise against you using. Another, more legitimate use, is in delegating constructors; you can have one constructor that takes lots of different options that is really an implementation detail, so you make it private, but then your remaining constructors delegate to it.

As an example of delegating constructors, the following class allows you to save a value and a type, but it only lets you do it for a subset of types, so making the general constructor private is needed to ensure that only the permitted types are used. The common private constructor helps code reuse.

public class MyClass {
     private final String value;
     private final String type;

     public MyClass(int x){
         this(Integer.toString(x), "int");
     }

     public MyClass(boolean x){
         this(Boolean.toString(x), "boolean");
     }

     public String toString(){
         return value;
     }

     public String getType(){
         return type;
     }

     private MyClass(String value, String type){
         this.value = value;
         this.type = type;
     }
}

Edit
Looking at this answer from several years later, I would like to note that this answer is both incomplete and also a little bit extreme. Singletons are indeed an anti-pattern and should generally be avoided where possible; however, there are many uses of private constructors besides singletons, and my answer names only one.

To give a couple more cases where private constructors are used:

1. To create an uninstantiable class that is just a collection of related static functions (this is basically a singleton, but if it is stateless and the static functions operate strictly on the parameters rather than on class state, this is not as unreasonable an approach as my earlier self would seem to suggest, though using an interface that is dependency injected often makes it easier to maintain the API when the implementation requires larger numbers of dependencies or other forms of context).

2. When there are multiple different ways to create the object, a private constructor may make it easier to understand the different ways of constructing it (e.g., which is more readable to you new ArrayList(5) or ArrayList.createWithCapacity(5), ArrayList.createWithContents(5), ArrayList.createWithInitialSize(5)). In other words, a private constructor allows you to provide factory function’s whose names are more understandable, and then making the constructor private ensures that people use only the more self-evident names. This is also commonly used with the builder pattern. For example:

   MyClass myVar = MyClass
       .newBuilder()
       .setOption1(option1)
       .setOption2(option2)
       .build();