Catch all standard exceptions in C API. (#6220)

* `std::bad_alloc` is not guaranteed to be caught.

src/c_api/c_api_error.h

@@ -16,9 +16,17 @@
 #else  // LOG_CAPI_INVOCATION
 #define API_BEGIN() try {
 #endif  // LOG_CAPI_INVOCATION
+
 /*! \brief every function starts with API_BEGIN();
      and finishes with API_END() or API_END_HANDLE_ERROR */
-#define API_END() } catch(dmlc::Error &_except_) { return XGBAPIHandleException(_except_); } return 0;  // NOLINT(*)
+#define API_END()                                                              \
+  } catch (dmlc::Error & _except_) {                                           \
+    return XGBAPIHandleException(_except_);                                    \
+  } catch (std::exception const &_except_) {                                   \
+    return XGBAPIHandleException(dmlc::Error(_except_.what()));                \
+  }                                                                            \
+  return 0; // NOLINT(*)
+
 #define CHECK_HANDLE() if (handle == nullptr) \
   LOG(FATAL) << "DMatrix/Booster has not been intialized or has already been disposed.";
 /*!