Yes it is possible to make a personal IVR system. The Cortana Intelligence Suite provides advanced analytics tools backed by Microsoft's Azure cloud platforms—data ingestion, data storage, data processing and advanced analytics components—all of the essential elements for building an automated IVR solution. It is deployed using the fully integrated service stack that manages speech requests via Skype and the Microsoft Bot Framework. You will need an Azure subscription.
Here's a short description of how the system works
AI and NLP Azure Services
Bing Speech transcribes raw speech data into text form
LUIS (Language Understanding Intelligent Service) identifies caller intent (place an order, find an order, etc.) and spoken entities (product categories, dates, times, etc.) from the transcribed text.
Azure Cognitive Search matches and ranks products from the inventory against the caller's description and applies advanced text analysis for common synonyms.
Microsoft Bot Framework processes conversation results from the caller and manages the call workflow.
These are the supporting Azure Services
Azure Web App hosts the core logic of the bot as a Node.js web app.
Cosmos DB stores shared session state for every conversation, allowing the web application to scale out in a stateless architecture.
Azure SQL stores product inventory and order status.