The Kalman filter is one alternative to fill in missing observations in time series. See this post as an example. The Kalman filter is a common algorithm that will be available in most languages and statistical software. Contrary to the Holt-Winters filter you have to specify a model for the data.
"How many consecutive gaps may make data set invalid for forecasting?
How many total gaps in data set makes it as invalid."
I don't know a rule to measure this. I would say it depends on how much
we know about the data and their context. Forecasting and, in general, the analysis of data involve a combination of our knowledge or theories and statistical methods to test our theories or find some further facts that we may have overlooked.
The amount of data or the presence of gaps may or may not be critical. For example, I have not looked at historical data about temperatures recorded in my town but I would be quite confident to give you a relatively narrow interval
about the temperatures that will be observed in the next days. On the other hand, I have a data base with thousands of flight prices and at this moment
I wouldn't dare to tell you whether you should buy a ticket today or wait
So there is a combination of knowledge and data. On one side, we may know a lot about the data but we lack a minimal amount of data. On the other side, we may have a huge amount of data but they don't have much meaning to us. In the former case, we may decide to throw the data away and trust our expert knowledge to foresee the future. In the latter case, we may throw the data into a brute force algorithm (some kind of machine learning algorithm) and let it find patterns and forecasts for us.
Usually we are at some point in-between these extreme cases. You are the one who knows how much the available data can contribute to your knowledge and how much uncertainty will be in the forecasts.