Eventually it is a numbers game. It is likely cheaper to make large numbers of relatively fast offensive missiles than it is to make an equal number of fast AND exceptionally accurate defensive missiles. This is why the US Department of Defense is going in so heavily on microwave and laser technologies. By layering multiple overlapping technologies like this (along with upgraded satellites to pinpoint launches as they happen and help determine the target), you have a much higher likelihood of success. However, if it was just missiles vs missiles, the aggressive side always has a better chance. They just need to have 1-2 successes to cause mayhem. The defender has to have 100% success to prevent mayhem.

It's not impossible, just not economical. Terminal intercept ABM systems like Aegis only protect a small area (10s to 100s of km radius) around the missile system - perfectly adequate for the defense of a carrier group which is their primary purpose. The missiles they are defending against have a range of many thousands of kilometers, or in the case of ICBMs can hit anywhere in the world, and thus one missile might threaten many thousands of potential targets. If your adversary is a poor pariah state only interested in a limited region, then making 10 interceptors for every one of their missiles might be viable, but if your adversary is a major power with a strong industrial base, they can easily build more missiles than you can build interceptors to defend against.

For a truly effective ballistic missile defense, you need a theater-level interceptor. The US is also actively working on this in the form of its Ground Based Mid-Course Defense (GMD). Nominally such a system can defend large sections of a continent and thus the cost of defense would be comparable to the cost of offense, but in practice mid-course intercept is much harder - you need a much more powerful missile and the adversary can use decoys and other counter measures. The current GMD program is only aimed at defeating rogue states like North Korea because their countermeasure capabilities are presumed to lag substantially behind the major powers. While it is not inconceivable that GMD's technical hurdles will one day be overcome, this system would not defend against hypersonic glide vehicles already under development by the major powers.

The history of warfare is a constant race between advancements in weapons and defenses. It is reasonably safe to assume that in the 40 some odd years it would take to develop an effective defense against any new delivery system, a further new delivery system can also be developed. Even if defense catches up for a bit, it will be momentary.

It's pretty much established that usable anti-ballistic missile defence against massive attack is impossible when you the counter-strategies.

If you spend trillon to build the system, the enemy counters it using just hundred billion or so.